8 short answer calc questions

1.

Write the limit as n goes to infinity of the summation from k equals 1 of the product of the 3rd power of the quantity 2 plus 5 times k over n and 5 over n as a definite integral. (10 points)

2.

Find the derivative with respect to x of the integral from 2 to x cubed of the natural log of x squared, dx . (10 points)


3.

Find the area bounded by the curves y2 = 2x + 6 and x = y + 1. Your work must include an integral in one variable. (10 points)


4.

Write the integral in one variable to find the volume of the solid obtained by rotating the first-quadrant region bounded by y = 0.5x2 and y = x about the line x = 3. (10 points)


5.

A rock is thrown upward with a speed of 48 feet per second from the edge of a cliff 400 feet above the ground. What is the speed of the rock when it hits the ground? Use acceleration due to gravity as -32 feet per second squared and approximate your answer to 3 decimal places. (10 points)


6.

Use the mid-point rule with n = 4 to approximate the area of the region bounded by y = x3 and y = x. (10 points)


7.

Electricity is consumed between the hours of midnight (t = 0) and 6 a.m. (t = 6). Selected values of the rate of consumption are shown in the table below, with t measured in hours past midnight, and R(t) measured in KW per hour. Use 4 trapezoids to estimate the total amount of electricity to the nearest KW consumed between midnight and 6 a.m. (10 points)

t (hours) 0 1 2 5 6
R(t) KW/hr 400 200 350 200 150


8.

Show that the function  F of x equals the integral from x to 4 times x of 1 over t dt is constant on the interval (0, +∞). (10 points)

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