Study Guide Cal2

Find the area enclosed by the given curves.

1) y = 2x - x2, y = 2x - 4

1) 32/3

Find the area enclosed by the given curves.

2) y = x3, y = 4x

8

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.

3) y = x + 3, y = 0, x = -3, x = 6

243π

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.

4) y = x + 2, y = 0, x = -2, x = 5

343/3 π

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.

5) y = x2 + 2, y = 0, x = 0, x = 3

243 /5 π

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.

6) y = x2 - 1, y = 0, x = 0, x = 4

1024 / 5 π

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.

7) y = 2x, y = 2, x = 0

8 / 3 π

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis.

8) y = 8x, y = 8, x = 0

128 / 3 π

Find the volume of the solid generated by revolving the region about the given line.

9) The region bounded above by the line y = 4, below by the curve y = 4 - x2, and on the right by the line x = 2, about the line y = 4

32 / 5 π

Find the volume of the solid generated by revolving the region about the given line.

10) The region bounded above by the line y = 16, below by the curve y = 16 - x2, and on the right by the line x = 4, about the line y = 16

1024 / 5 π

Find the volume of the solid generated by revolving the region about the y-axis.

11) The region enclosed by x = y2 4 , x = 0, y = - 4, y = 4

128 / 5 π

Find the volume of the solid generated by revolving the region about the y-axis.

12) The region enclosed by x = y2 2 , x = 0, y = - 2, y = 2

16 / 5 π

Set up an integral for the length of the curve.

13) y = 4 cos x, 0 ≤ x ≤ π

π ∫0 1 + 16 sin2 ∫ x dx

Set up an integral for the length of the curve.

14) y = x5, 0 ≤ x ≤ 1

1∫ 0 1 + 25x8 ∫ dx

Write the integral that gives the surface area generated when the curve is revolved about the x-axis. Do not simplify

15) y = x9 on [0, 4]

2π 4 ∫0 x9 1 + (9x8) 2 ∫ dx

Write the integral that gives the surface area generated when the curve is revolved about the x-axis. Do not simplify

16) y = tanx on {0, π/4}

2π π/4∫ 0 tanx 1 + sec4x dx

Evaluate the integral using integration by parts.

17) ∫ -8x cos 6x dx

- 2/9 cos 6x - 4/3 x sin 6x + C

Evaluate the integral.

18) ∫ 24x sin x dx

24 sin x - 24x cos x + C

Evaluate the integral.

19) 9xe^x ∫ dx

9xex - 9ex + C

Evaluate the integral.

20) e2x x2 ∫ dx

1/2x2e2x - 1/2 xe2x + 1/4 e2x + C

Evaluate the integral.

21) x3 ∫ ln 9x dx

1/4 x4 ln 9x - 1/16 x4 + C

Use trig integration to evaluate the integral.

22) 2 cos3 x sin7 ∫ x dx

1/4 sin8 x - 1/5 sin10 x + C

Use trig integration to evaluate the integral.

23) 2 cos3 x sin6 ∫ x dx

2/7 sin7 x - 2/9 sin9 x + C

Integrate the function using trig substitutions. (Number 24-25)

24) 1 /t2 |^9 - t2 ∫ dt

- /^9 - t2 / 9t + C

Integrate the function using trig substitutions. (Number 24-25)

25) x3/ |^x2 + 7 ∫ dx

1/3 (x2 + 7) 3/2 - 7 /^x2 + 7 + C

Give the appropriate form of the partial fraction decomposition.

26) 3x + 29 / (x + 3)(x + 7)

5 / x + 3 + -2 / x + 7

Give the appropriate form of the partial fraction decomposition.

27) x + 6 / (x + 2) 2

1/ x + 2 + 4/ (x + 2)2

Express the integrand as a sum of partial fractions and evaluate the integral

28) 2x + 32 / x2 + 8x + 7 ∫ dx

ln | (x + 1)5 / (x + 7)3 | + C

Express the integrand as a sum of partial fractions and evaluate the integral

29) x3 / x2 + 8x + 16 ∫ dx

x2/2 - 8x + 48 ln|x + 4| + 64 / x + 4 + C

Express the integrand as a sum of partial fractions and evaluate the integral

30) 6x2 + x + 45 / x3 + 9x ∫ dx

5 ln |x| + 1/2 ln |x2 + 9| + 1/3 tan-1 (x /3 )+ C

Express the integrand as a sum of partial fractions and evaluate the integral

31) 4x2 + x + 108 / x3 + 36x ∫ dx

3 ln |x| + 1/2 ln |x2 + 36| + 1/6 tan-1 (x/6) + C

Evaluate the integral

32) x4 - 1 / 9 + x2 ∫ dx

1/3 x3 - 9x + 80/3 tan-1 (x/3)+ C