91Ó°ÊÓ

In Exercises \(1-14\), evaluate the given integral. $$\int_{-1}^{2}\left(\frac{x^{2}}{3}-\frac{2}{9} x\right) d x$$

Determine \(\Delta x\) and the midpoints of the subintervals formed by partitioning the given interval into \(n\) subintervals. $$0 \leq x \leq 2 ; n=4$$

Find the value of \(k\) that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How?] $$\int(5 x-7)^{-2} d x=k(5 x-7)^{-1}+C$$

Find the volume of the solid of revolution generated by revolving about the \(x\)-axis the region under each of the following curves. \(y=x^{2}\) from \(x=1\) to \(x=2\)

Find the volume of the solid of revolution generated by revolving about the \(x\)-axis the region under each of the following curves. \(y=\sqrt{x}\) from \(x=0\) to \(x=4\) (The solid generated is called a paraboloid.)

Find the value of \(k\) that makes the antidifferentiation formula true. [Note: You can check your answer without looking in the answer section. How?] $$\int(3 x+2)^{4} d x=k(3 x+2)^{5}+C$$

Use a Riemann sum with \(n=4\) and right endpoints to estimate the area under the graph of \(f(x)=2 x-4\) on the interval \(2 \leq x \leq 3 .\) Then, repeat with \(n=4\) and midpoints. Compare the answers with the exact answer, \(1,\) which can be computed from the formula for the area of a triangle.

Displacement versus Distance Traveled The velocity of an object moving along a line is given by \(v(t)=2 t^{2}-3 t+1\) feet per second. (a) Find the displacement of the object as \(t\) varies in the interval \(0 \leq t \leq 3\) (b) Find the total distance traveled by the object during the interval of time \(0 \leq t \leq 3\)

A ball is thrown upward from a height of 256 feet above the ground, with an initial velocity of 96 feet per second. From physics it is known that the velocity at time \(t\) is \(v(t)=96-32 t\) feet per second. (a) Find \(s(t),\) the function giving the height above the ground of the ball at time \(t\) (b) How long will the ball take to reach the ground? (c) How high will the ball go?

A rock is dropped from the top of a 400 -foot cliff. Its velocity at time \(t\) seconds is \(v(t)=-32 t\) feet per second. (a) Find \(s(t),\) the height of the rock above the ground at time \(t\) (b) How long will the rock take to reach the ground? (c) What will be its velocity when it hits the ground?