# Exam 3 Study Guide

1)

2)

∫

3)

4)

∫ sin θ(cot θ + csc θ) dθ

4)

Use a ﬁnite approximation to estimate the area under the graph of the given function on the stated interval as instructed. 5) f(x) = x2 between x = 2 and x = 6 using the "midpoint rule" with four rectangles of equal 5) width. Graph the function f(x) over the given interval. Partition the interval into 4 subintervals of equal length. Then add to 4 your sketch the rectangles associated with the Riemann sum ∑ f(ck) Δxk , using the indicated point in the kth k=1 subinterval for ck. 6) f(x) = x2 - 1, [0, 8], right-hand endpoint 6) y 56 52 48 44 40 36 32 28 24 20 16 12 8 4 -4 2 4 6 x

Find the formula and limit as requested. 7) For the function f(x) = 2x2+ 2, ﬁnd a formula for the upper sum obtained by dividing the interval [0, 3] into n equal subintervals. Then take the limit as n→∞ to calculate the area under the curve over [0, 3].

7)

1

Evaluate the integral. 0 8) 3x2 + x + 3 dx

∫

8)

6 Find the derivative. x3 sin t dt 9) d dx 0

∫

9)

Find the total area of the region between the curve and the x-axis. 10) y = x2(x - 2)2; 0 ≤ x ≤ 2 Find the area of the shaded region. 11)

10)

11)

Evaluate the integral. 12)

∫ x2 ∫

x3 + 3 dx

12)

13)

sin t dt (8 + cos t)6

13)

Solve the problem. 14) Given the acceleration, initial velocity, and initial position of a body moving along a coordinate line at time t, ﬁnd the body's position at time t. a = 32 cos 4t, v(0) = -10, s(0) = 12 Use the substitution formula to evaluate the integral. 4 9- x 15) dx x 1

14)

∫ ∫

15)

16)

π/2

cot x csc3 x dx

16)

π/6

2

Find the area of the shaded region. 17)

y 25 20 15 10 5 -5 -4 -3 -2 -1 -5 -10 -15 -20 (-4, -24)-25 (0, 0) 1 2 3

f(x) = x3 + x2 - 6x g(x) = 6x

(3, 18)

17)

4

5

x

Find the area enclosed by the given curves. 18) y = 1 x 2, y = -x2 + 6 2 19) Find the area of the region in the ﬁrst quadrant bounded by the line y = 8x, the line x = 1, the curve y = 1 , and the x-axis. x

18)

19)

Find the volume of the solid generated by revolving the region bounded by the given lines and curves about the x-axis. 20) y = 2x + 3, y = 0, x = 0, x = 1 20) 21)

21) y = x2 + 3, y = 3x + 3 Find the volume of the solid generated by revolving the region about the given line. 22) The region in the ﬁrst quadrant bounded above by the line y = 3, below by the curve y = 3x, and on the left by the y-axis, about the line x = -1 23) The region in the ﬁrst quadrant bounded above by the line y = 6x3, below by x-axis, and on the right by the line x = 1, about the line y = - 1 Find the volume of the solid generated by revolving the region about the y-axis. 2 24) The region enclosed by x = y , x = 0, y = - 4, y = 4 4

22)

23)

24)

3

Answer Key Testname: EXAM 3 PRACTICE

1) - 1 + 1 x1/2 2x2 2 2) 1 sin πx - 36 cos x π 6 6 3) 2 t3/2 - t7/6 + C 7 3 4) sin θ + θ + C 5) 69 6) y 56 52 48 44 40 36 32 28 24 20 16 12 8 4 -4 2 4 6 x

3 2 7) 6 + 108n + 162n + 54n ; Area = 24 6n3 8) - 252 9) 3x2 sin (x3) 10) 16 15 11) 36 12) 2 x3 + 3 3/2 + C 9 13) 1 +C 5(8 + cos t)5

14) s = - 2 cos 4t - 10t + 12 15) 15 16) 7 3 17) 937 12 18) 16 19) 5 4 20) 4π 4

Answer Key Testname: EXAM 3 PRACTICE

21) 297 π 5 22) 57 π 5 23) 57 π 7 24) 128 π 5

5

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