# Question 1: (a) (8 pts) Estimate the area under the graph of f (x)...

## Transcribed Text

Question 1: (a) (8 pts) Estimate the area under the graph of f (x) = 9 x2 on the interval [3, 3] using 3 approximating rectangles and the right end points. (b) (2 pts) Find an antiderivative of f(x) = 9 x2. (c) (8 pts) Using part 2 of the Fundamental Theorem of Calculus find the area under the graph of f(x)=9x2 fromx=3tox=3. 2 Question 2: (a) (8 pts) Express the Riemann sum n!1 i=1 n n as a definite integral on the interval [1, 1]. (b) (8 pts) Express the definite integral Z 12 px dx as the limit of a Riemann sum. Xn 2 ✓ 2i◆ lim sin 1+ 5 3 Question 3: (a) (8 pts) Use FTC part 1 to find the derivative of g(x) = tan t p 3 dt Z sinx 8 1+t (b) (8 pts) Evaluate the indefinite integral Z✓ 4 +sec2x+1◆dx 1+x2 x 4 Question 4: (a) (8 pts) Use u-substitution to evaluate the following indefinite integral: Z 4 sin(ln x) dx x (b) (8 pts) Use integration by parts to evaluate the following indefinite integral: Z 3xe8x dx 5 Question 5: Consider the region R bounded by x = 5 y2 and x = 14 y2. (a) (6 pts) Graph x = 5 y2 and x = 14 y2, then identify the region bounded by the equations. Clearly label the graph. (b) (4 pts) For what values of y does 5y2 = 14y2? y (c) (6 pts) Find the area of the region R bounded by the functions x = 5 y2 and x = 14y2. 6 x Question 6: (a) (8 pts) Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by y = ex, y = 0, x = 0, and x = 1 about the x-axis. (b) (8 pts) Set up, but do not evaluate, an integral for the volume of the solid obtained by rotating the region bounded by y = ex, y = 0, x = 0, and x = 1 about the line x = 2. 7

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