Exam 6: The Definite Integral

Find all antiderivatives of the function. -f(x) = $\mathrm { e } ^ { - \mathrm { x } / 2 }$ Enter your answer with any fractional coefficients and powers in reduced form $\frac { a } { b }$ .

In the figure below, the region enclosed by the curves y = - $\frac { 4 } { x }$ , y = -x, and y = -x + 3 is shown. Set up an integral or sum of integrals to find the area of the shaded region. (Do not calculate the area.)

Find the area of the region bounded by y = 4x - $x ^ { 2 }$ and the x-axis. Enter a reduced fraction of form $\frac { a } { b }$ .

$\int _ { 0 } ^ { 2 } \left( 3 e ^ { 4 - 2 x } \right) d x$ Enter your answer as a(b + $\mathrm { e } ^ { \mathrm { C } }$ ) with any fractions in reduced form $\frac { e } { f }$ .

What is the area under the curve y = $x ^ { 3 }$ + x from x = 1 to x = 2?

Find: $\int$ $\left( \frac { 6 } { 5 } x ^ { 5 } + 4 e ^ { - 2 x } \right)$ dx

A region is bounded above by the graph of y = $x ^ { - 2 }$ and below by the x-axis on the interval from $x = 1$ to x = 3. Find the volume of the solid of revolution generated by revolving the region about the x-axis. Enter your answer as a reduced quotient of form $\frac { a b } { c }$ .

Find: $\int \frac { d x } { x ^ { 1 / 5 } }$ Enter your answer as a power function in x in standard form with any fractional coefficients or powers in reduced form $\frac { a } { b }$ and any constant at the right end.

A region is bounded by the graph of y = $x ^ { 3 }$ , the y-axis, and the horizontal line $y = 1$ . Find the volume of the solid of revolution generated by revolving the region about the x-axis. Enter your answer as a reduced quotient in form $\frac { a b } { c }$ .

Find: $\int ( 3 x + 2 ) ^ { 2 } d x$ Enter your answer as a polynomial in x in standard form with any fractional coefficients or powers reduced of form $\frac { a } { b }$ .

Find the area of the region bounded by y =1 - 2x - $x ^ { 2 }$ and the lines x = -1 and x = 0. Enter a reduced fraction of form $\frac { a } { b }$ .

Find the area of the region bounded by the curve f(x) = 5x - 2 $x ^ { 2 }$ and the line y = 3. Enter just a reduced fraction of form $\frac { a } { b }$ .

$\int _ { 1 } ^ { 2 } 5 x d x$ Enter your answer as a reduced fraction of form $\frac { a } { b }$ .

Suppose that the marginal revenue for a retailer is 6 $x ^ { 2 }$ - $\sqrt { x }$ + x dollars at sales level x. If 4 units are currently being sold, what is the extra revenue received from the sale of 5 additional units?

A helicopter rises straight up in the air so that its velocity t seconds after take-off is $\mathrm { v } ( \mathrm { t } ) = \mathrm { t } ^ { 3 / 2 } + \frac { 1 } { 2 } \mathrm { t } ^ { 1 / 2 } + 1$ feet per second. If the landing pad is 100 feet above the ground, which of the following gives the height of the helicopter at time t ?