Chapter 7: Definite Integrals

301. BC: Evaluate \( \sum_{i=1}^5 i^3 \).

解答

答案： ( D )

\( 1^3+2^3+3^3+4^3+5^3 = 225 \); or \( \frac{n^2(n+1)^2}{4} = \frac{25\cdot36}{4} = 225 \) .

302. Evaluate \( \int_0^4 (x^3 - 3x + 1)\,dx \).

解答

答案： ( D )

\( \left[\frac{x^4}{4}-\frac{3x^2}{2}+x\right]_0^4 = 64-24+4-0 = 44 \) .

303. Evaluate \( \int_{\pi/2}^{\pi} \frac{1}{4+x^2} dx \).

解答

答案： ( A )

\( \frac{1}{2}\tan^{-1}(x/2)\big|_{\pi/2}^{\pi} \approx 0.502 - 0.333 \approx 0.169 \) .

304. BC: Evaluate \( \int_0^4 \frac{1}{\sqrt{16-x^2}} dx \).

解答

答案： ( C )

Improper integral; limit as upper bound \( \to 4^- \) gives \( \sin^{-1}(x/4)\big|_0^4 = \frac{\pi}{2} \) .

305. Evaluate \( \int_0^{2\pi} (x - \cos x)\,dx \).

解答

答案： ( D )

\( \left[\frac{x^2}{2}-\sin x\right]_0^{2\pi} = 2\pi^2 \) .

306. If \( \int_0^k (5-x)\,dx = -12 \) and \( k>0 \), find \( k \).

解答

答案： ( D )

\( 5k - \frac{k^2}{2} = -12 \) gives \( k^2-10k-24=0 \); \( (k+2)(k-12)=0 \); \( k=12 \) .

307. BC: Evaluate \( \sum_{i=1}^n n^2 i(i+2) \).

解答

答案： ( A )

\( n^2\sum(i^2+2i) = n^3(n+1)[\frac{1}{3}n+\frac{7}{6}] \) .

308. Find \( \int_{-2}^4 \frac{|x|}{2x} dx \).

解答

答案： ( B )

Split: \( \frac{|x|}{2x} = -\frac{1}{2} \) on \( [-2,0) \) and \( \frac{1}{2} \) on \( (0,4] \); net area = 1 .

309. Find \( \frac{dy}{dx} \) if \( y = \int_x^{x^2} (t^2-t+1)\,dt \).

解答

答案： ( B )

Evaluate integral then differentiate; \( \frac{dy}{dx} = 2x^5-2x^3-x^2+3x-1 \) .

310. Evaluate \( \int_0^6 |2x-4|\,dx \).

解答

答案： ( B )

Split at \( x=2 \); \( \int_0^2 (4-2x)\,dx + \int_2^6 (2x-4)\,dx = 4 + 16 = 20 \) .

311. Evaluate \( \int_0^{\pi/2} \frac{\sin x}{\cos x+1} dx \).

解答

答案： ( C )

Let \( u=\cos x+1 \), \( du=-\sin x\,dx \); \( -\int \frac{du}{u} \) from 1 to 0 gives \( \ln 2 \approx 0.693 \) .

312. BC: Evaluate \( \int_{-\infty}^0 e^x dx \).

解答

答案： ( B )

\( \lim_{k\to-\infty}\int_k^0 e^x dx = 1 - e^k \to 1 \) .

313. Find \( k \) if \( \int_{-4}^k (2x-3)\,dx = -30 \) and \( k>1 \).

解答

答案： ( B )

\( [x^2-3x]_{-4}^k = k^2-3k-28 = -30 \); \( k^2-3k+2=0 \); \( k=2 \) .

314. Find \( \frac{dy}{dx} \) if \( y = \int_x^{x^2} \frac{1}{\sqrt{t}} dt \).

解答

答案： ( C )

\( y = 2\sqrt{x^2}-2\sqrt{x} = 2x-2\sqrt{x} \) for \( x>0 \); \( \frac{dy}{dx} = 2 - \frac{1}{\sqrt{x}} \) .

315. Evaluate \( \int_{-2}^2 (x^2-2x+1)\,dx \).

解答

答案： ( D )

\( \left[\frac{x^3}{3}-x^2+x\right]_{-2}^2 = \frac{16}{3}+\frac{12}{3} = 9\frac{1}{3} \) .

316. BC: Evaluate \( \sum_{i=0}^6 i^2(i-1) \).

解答

答案： ( D )

\( \sum i^3 - \sum i^2 = \frac{36\cdot49}{4} - \frac{6\cdot7\cdot13}{6} = 441-91 = 350 \) .

317. Evaluate \( \int_0^{\pi/2} \sin^2 x\,dx \).

解答

答案： ( C )

\( \int_0^{\pi/2} \frac{1-\cos 2x}{2}\,dx = \left[\frac{x}{2}-\frac{\sin 2x}{4}\right]_0^{\pi/2} = \frac{\pi}{4} \) .

318. Evaluate \( \int_0^1 x(x+1)^{1/3} dx \).

解答

答案： ( D )

Let \( u=x+1 \); \( \int (u-1)u^{1/3}\,du \) from 1 to 2; evaluate \( \approx 0.59 \) .

319. Evaluate \( \int_{-6}^6 |x^2-9|\,dx \).

解答

答案： ( C )

Split at \( x=\pm 3 \); symmetric pieces sum to 108 .

320. Evaluate \( \int_1^4 \frac{e^{\sqrt{x}}}{\sqrt{x}} dx \).

解答

答案： ( A )

Let \( u=\sqrt{x} \); \( 2\int_1^2 e^u du = 2e^u\big|_1^2 = 2e(e-1) \) .

321. Evaluate \( \int_{\ln 2}^{\ln 3} x e^{x^2} dx \).

解答

答案： ( D )

Let \( u=x^2 \); \( \frac{1}{2}\int e^u du \); \( \frac{1}{2}e^{x^2}\big|_{\ln 2}^{\ln 3} = \frac{9}{2}-\frac{4}{2} = 2.5 \) .

322. Evaluate \( \int_0^3 |4x-2|\,dx \).

解答

答案： ( C )

Split at \( x=1/2 \); \( \int_0^{1/2}(2-4x)\,dx + \int_{1/2}^3 (4x-2)\,dx = 13 \) .

323. Evaluate \( \int_0^{\pi/3} \tan^2 x\sec^2 x\,dx \).

解答

答案： ( C )

Let \( u=\tan x \); \( \frac{1}{3}\tan^3 x\big|_0^{\pi/3} = \frac{(\sqrt{3})^3}{3} = \sqrt{3} \) .

324. Find \( k \) if \( \int_0^k (2x-\frac{\sqrt{x}}{2})\,dx = 0 \) and \( k>0 \).

解答

答案： ( A )

\( k^2 - \frac{1}{3}k^{3/2} = 0 \); \( k^{1/2}(k^{1/2}-\frac{1}{3})=0 \); \( k = \frac{1}{9} \) .

325. BC: Evaluate \( \int_2^3 \frac{1}{\sqrt{x-2}} dx \).

解答

答案： ( D )

Improper; \( 2\sqrt{x-2}\big|_2^3 = 2\sqrt{1} = 2 \) .

326. Evaluate \( \int_0^1 10^{3x} dx \).

解答

答案： ( B )

Let \( u=3x \); \( \frac{10^{3x}}{3\ln 10}\big|_0^1 = \frac{1000}{3\ln 10}-\frac{1}{3\ln 10} \approx 144.62 \) .

327. Evaluate \( \int_e^{e^2} \frac{1}{x\ln x} dx \).

解答

答案： ( D )

Let \( u=\ln x \); \( \ln|\ln x|\big|_e^{e^2} = \ln 2 - \ln 1 = \ln 2 \) .

328. Find the average value of \( f(x)=\frac{|x|}{2x} \) on \( [-2,4] \).

解答

答案： ( A )

\( \frac{1}{6}\int_{-2}^4 \frac{|x|}{2x}\,dx \); split and evaluate gives \( \frac{1}{6} \) .

329. Evaluate \( \int_4^9 \frac{x+1}{\sqrt{x}} dx \).

解答

答案： ( C )

\( \int (x^{1/2}+x^{-1/2})\,dx = \frac{2}{3}x^{3/2}+2x^{1/2} \); evaluate from 4 to 9 gives \( 14\frac{2}{3} \) .

330. BC: Evaluate \( \int_{-\infty}^{\infty} \frac{1}{1+x^2} dx \).

解答

答案： ( D )

\( \int_{-\infty}^0 + \int_0^{\infty} = \frac{\pi}{2}+\frac{\pi}{2} = \pi \) .

331. If \( f(x)=\int_{\pi/2}^x \tan^{-1} t\,dt \), find \( f'(\pi/6) \) to the nearest thousandth.

解答

答案： ( B )

FTC: \( f'(x)=\tan^{-1} x \); \( f'(\pi/6)=\tan^{-1}(\pi/6)\approx 0.4823 \) .

332. Evaluate \( \int_{-1}^1 (x^2-3)(x^5+2)\,dx \).

解答

答案： ( C )

Expand; odd powers cancel; \( \int_{-1}^1 (x^7-3x^5+2x^2-6)\,dx = -10\frac{2}{3} \) .

333. BC: Evaluate \( \int_3^5 \frac{1}{x^2-3x+2} dx \).

解答

答案： ( D )

Partial fractions \( \frac{1}{(x-2)(x-1)} \); \( \ln\frac{|x-2|}{|x-1|}\big|_3^5 = \ln\frac{3}{4}-\ln\frac{1}{2} = \ln\frac{3}{2} \approx 0.405 \) .

334. Evaluate \( \int_0^1 \frac{x^2}{\sqrt{1-x^6}} dx \).

解答

答案： ( C )

Let \( u=x^3 \); \( \frac{1}{3}\sin^{-1}(x^3)\big|_0^1 = \frac{1}{3}\cdot\frac{\pi}{2} = \frac{\pi}{6} \) .

335. Find \( \frac{dy}{dx} \) if \( y = \int_{\sin x}^{\cos x} (1-\frac{1}{2}t)\,dt \).

解答

答案： ( A )

FTC and chain rule: \( (1-\frac{1}{2}\cos x)(-\sin x) - (1-\frac{1}{2}\sin x)(\cos x) = \sin x\cos x - \sin x - \cos x \) .

336. Evaluate \( \int_{\pi/6}^{\pi/4} \csc^2 x\cot x\,dx \).

解答

答案： ( D )

Let \( u=\cot x \); \( -\frac{\cot^2 x}{2}\big|_{\pi/6}^{\pi/4} = -\frac{1}{2}+\frac{3}{2} = 1 \) .

337. Evaluate \( \int_1^e \frac{\ln x}{5x} dx \).

解答

答案： ( D )

Let \( u=\ln x \); \( \frac{1}{10}(\ln x)^2\big|_1^e = \frac{1}{10} \) .

338. On \( [-2,4] \) there must be a value \( c \) with \( f(c)=\frac{1}{6} \) by the Mean Value Theorem for Integrals. What does \( c \) equal?

解答

答案： ( D )

MVT for integrals requires continuity; \( f(x)=\frac{|x|}{2x} \) has a discontinuity at \( x=0 \) .

339. If \( f'(x)=g(x) \), express \( \int_0^\pi g(2x)\,dx \) in terms of \( f(x) \).

解答

答案： ( D )

Substitute \( u=2x \); \( \frac{1}{2}\int_0^{2\pi} g(u)\,du = \frac{1}{2}[f(2\pi)-f(0)] \) .

340. Evaluate \( \int_{\ln 3}^{\ln 5} \frac{e^x}{e^x+4} dx \).

解答

答案： ( A )

Let \( u=e^x+4 \); \( \ln|e^x+4|\big|_{\ln 3}^{\ln 5} = \ln 9 - \ln 7 = \ln\frac{9}{7} \) .

341. Evaluate \( \int_0^{\pi/6} \sqrt{\sin x}\,\cos x\,dx \).

解答

答案： ( C )

Let \( u=\sin x \), \( du=\cos x\,dx \); \( \int u^{1/2}du = \frac{2u^{3/2}}{3}+C \); \( \frac{2(\sin x)^{3/2}}{3}\big|_0^{\pi/6} = \frac{2(1/2)^{3/2}}{3} = \frac{\sqrt{2}}{6} \) .

342. If \( G(x) \) is the antiderivative of \( \ln(x) \) and \( G(1)=0 \), find \( G(2) \).

解答

答案： ( C )

\( G(x)=\int\ln x\,dx = x\ln x - x + C \); \( G(1)=0 \Rightarrow C=1 \); \( G(2)=2\ln 2-2+1=2\ln 2-1 \) .

343. BC: Evaluate \( \int_0^{\pi/2} e^x\sin x\,dx \).

解答

答案： ( B )

Integration by parts twice: \( \int e^x\sin x\,dx = \frac{1}{2}e^x(\sin x-\cos x)+C \); evaluate from 0 to \( \pi/2 \) gives \( \frac{1}{2}(e^{\pi/2}+1) \) .

344. BC: Evaluate \( \int_6^{10} \frac{1}{x^2-3x-10}\,dx \).

解答

答案： ( B )

Partial fractions \( \frac{1}{(x-5)(x+2)} = \frac{1/7}{x-5}-\frac{1/7}{x+2} \); \( \frac{1}{7}[\ln|x-5|-\ln|x+2|]\big|_6^{10} = \frac{1}{7}\ln|\frac{10}{3}| \) .

345. BC: Evaluate \( \int_1^\infty \frac{1}{x^6}\,dx \).

解答

答案： ( B )

\( \lim_{k\to\infty}\int_1^k x^{-6}\,dx = \lim_{k\to\infty}[-\frac{x^{-5}}{5}]_1^k = \frac{1}{5} \) .

346A. 346. The marginal cost of producing \( x \) units is \( C'(x)=\frac{1}{4}x-2 \). (A) Find an expression for \( C(x) \), assuming the cost of producing 0 units is $2, so that \( C(0)=2 \).

解答

\( C(x)=\int C'(x)\,dx = \int(\frac{1}{4}x-2)\,dx = \frac{1}{8}x^2-2x+C \); \( C(0)=2 \Rightarrow C=2 \); \( C(x)=\frac{1}{8}x^2-2x+2 \) .

346B. 346. (B) Find the value of \( x \) such that the average cost is a minimum. Justify your answer.

解答

\( \bar{C}=\frac{C(x)}{x}=\frac{x}{8}-2+\frac{2}{x} \); \( \frac{d\bar{C}}{dx}=\frac{1}{8}-\frac{2}{x^2}=0 \) gives \( x=4 \); second derivative \( \frac{4}{x^3}>0 \) at \( x=4 \) so relative minimum; \( x=4 \) .

346C. 346. (C) Find the cost for producing 40 units.

解答

\( C(40)=\frac{40^2}{8}-2(40)+2=200-80+2=122 \) .

347A. 347. Area \( A \) is bounded by \( f(x)=x^2 \) and the \( x \)-axis. (A) Use a right-hand Riemann sum to find area \( A \) on the interval from \( x=0 \) to \( x=4 \), using 4 subdivisions of equal length.

解答

\( \Delta x_i=1 \); right-hand sum \( A = f(1)\cdot1+f(2)\cdot1+f(3)\cdot1+f(4)\cdot1 = 1+4+9+16=30 \) .

347B. 347. (B) Find the area \( A \) on the interval from \( x=0 \) to \( x=4 \) using 8 subdivisions of equal length.

解答

\( \Delta x_i=\frac{1}{2} \); right-hand sum \( A = \frac{1}{4}\cdot\frac{1}{2}+1\cdot\frac{1}{2}+\frac{9}{4}\cdot\frac{1}{2}+4\cdot\frac{1}{2}+\frac{25}{4}\cdot\frac{1}{2}+9\cdot\frac{1}{2}+\frac{49}{4}\cdot\frac{1}{2}+16\cdot\frac{1}{2} = \frac{204}{8}=25\frac{1}{2} \) .

347C. 347. (C) Find area \( A \) by integrating \( f(x) \) over the interval \( (0,4) \).

解答

\( A=\int_0^4 x^2\,dx = \frac{x^3}{3}\big|_0^4 = \frac{64}{3} = 21\frac{1}{3} \) .

348A. 348. A virus population grows at \( P'(t)=10t-2\sqrt{t}+100 \) organisms per hour. (A) If \( P(0)=500 \), what is the population after \( t \) hours?

解答

\( P(t)=\int(10t-2t^{1/2}+100)\,dt = 5t^2-\frac{4}{3}t^{3/2}+100t+C \); \( P(0)=500 \Rightarrow C=500 \); \( P(t)=5t^2-\frac{4}{3}t^{3/2}+100t+500 \) .

348B. 348. (B) What is the increase in population after 3 hours, rounded to the nearest whole number?

解答

\( P(3)-P(0)=5(9)-\frac{4}{3}(3\sqrt{3})+300 = 45-\frac{4\sqrt{27}}{3}+300 \approx 338 \) .

348C. 348. (C) What is the average rate of change in the population during the first 12 hours?

解答

\( P_{\text{avg}}=\frac{P(12)-P(0)}{12} = \frac{1}{12}[5(144)-\frac{4}{3}\sqrt{1728}+1200] \approx 155 \) per hour.

349A. 349. A particle starts at 100 on the \( x \)-axis with \( v(0)=50 \); acceleration \( a(t)=15\sqrt{t} \). (A) Find the velocity function.

解答

\( v(t)=\int a(t)\,dt = \int 15t^{1/2}\,dt = 10t^{3/2}+C \); \( v(0)=50 \Rightarrow C=50 \); \( v(t)=10\sqrt{t^3}+50 \) .

349B. 349. (B) Find the position function.

解答

\( s(t)=\int v(t)\,dt = 4t^{5/2}+50t+C \); \( s(0)=100 \Rightarrow C=100 \); \( s(t)=4t^{5/2}+50t+100 \) .

349C. 349. (C) What is the change in velocity between \( t=0 \) and \( t=5 \)?

解答

\( v(5)-v(0)=\int_0^5 a(t)\,dt = 10\sqrt{125}+50-50 = 50\sqrt{5} \approx 112 \) .

349D. 349. (D) What is the change in position from \( t=0 \) to \( t=5 \)?

解答

\( s(5)-s(0)=4(5^{5/2})+250+100-100 = 4\sqrt{3125}+250 \approx 474 \) .

350A. 350. A pillar is 35 ft tall; density \( \rho(x)=\frac{1}{3\sqrt{x+1}} \) (tons), \( x \) from ground. (A) What is the total mass of the pillar rounded to the nearest ton?

解答

\( m=\int_0^{35}\rho(x)\,dx = \frac{1}{3}\int_0^{35}(x+1)^{-1/2}\,dx = \frac{2}{3}(x+1)^{1/2}\big|_0^{35} = \frac{2}{3}(6-1)= \frac{10}{3} \approx 3 \) tons (nearest ton).

350B. 350. (B) For what value of height \( h \) does the interval \( [0,h] \) contain half the total mass?

解答

\( \int_0^h \rho(x)\,dx = \frac{1}{2}\cdot\frac{10}{3} \); \( \frac{2}{3}(h+1)^{1/2}-\frac{2}{3}=\frac{5}{3} \); \( (h+1)^{1/2}=\frac{7}{2} \); \( h+1=\frac{49}{4} \); \( h=\frac{45}{4}=11\frac{1}{4} \) ft.

350C. 350. (C) What is the mass of the uppermost 5 ft of the pillar, rounded to the nearest hundredth?

解答

\( m=\int_{30}^{35}\rho(x)\,dx = \frac{2}{3}(x+1)^{1/2}\big|_{30}^{35} = \frac{2}{3}(6-\sqrt{31}) \approx 0.29 \) tons.