Exam 6: The Definite Integral

Determine the average value of f(x) = $\frac { 1 } { x ^ { 2 } }$ over the interval from x = 1 to x = 2. Enter a reduced fraction of form $\frac { a } { b }$ .

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

Draw the region whose area is given by the definite integral. $\int _ { 8 } ^ { 9 } \sqrt [ 4 ] { x }$ dx

Use a Riemann sum to approximate the area under the graph of f(x) on the given interval. Use the right endpoints. Enter just a real number to two decimal places. f(x) = $x ^ { 3 }$ ; 0 ≤ x ≤ 2 , n = 4

Use a Riemann sum to approximate the area under the graph of $f ( x ) = x ^ { 3 } , 1 \leq x \leq 3 , n = 4$ Use the midpoints of the intervals. Enter your answer as just a reduced fraction of form $\frac { a } { b }$ .

What is the area under the curve f(x) = 3 $x ^ { 2 }$ + 2x + 1 from x = - $\frac { 3 } { 2 }$ to x = - $\frac { 1 } { 2 }$ ?

Find: $\int \left( 4 x ^ { 3 / 2 } - \frac { 1 } { 2 x ^ { 3 / 2 } } \right)$ dx Integrate the terms in the order they appear. Enter your answer as a sum of power functions in standard form with any fractional powers or coefficients reduced of form $\frac { a } { b }$ .

Given $f ( x ) = e ^ { x } + x ; 0 \leq x \leq 2 , n = 6 ,$ set up a Riemann sum to approximate the area under the graph of f(x) on the given interval. Use the left endpoints. Is the following the correct sum? $\frac { 1 } { 3 } \left[ 1 + \left( \mathrm { e } ^ { 1 / 3 } + \frac { 1 } { 3 } \right) + \left( \mathrm { e } ^ { 2 / 3 } + \frac { 2 } { 3 } \right) + ( \mathrm { e } + 1 ) + \left( \mathrm { e } ^ { 4 / 3 } + \frac { 4 } { 3 } \right) + \left( \mathrm { e } ^ { 5 / 3 } + \frac { 5 } { 3 } \right) \right]$

Find: $\int$ $\left( 4 \sqrt { x } - \frac { 1 } { 2 \sqrt { x } } \right)$ dx

What is the area under the curve y = $\frac { 1 } { 3 }$ + 3 $e ^ { 3 x }$ between x = - $\frac { 1 } { 3 }$ and x = $\frac { 1 } { 3 }$ ?

Calculate. - $\int _ { - 1 } ^ { 1 } e ^ { - 2 x } d x$

A ball is thrown upward with initial velocity of 144 feet per second. How high will the ball go? (Recall that from physics, it is known that the velocity at time t is $144 - 32 t$ feet per second.)Enter just an integer (no units).

$\int _ { 1 } ^ { 4 } 3 \sqrt { x } d x$ Enter just an integer.

A region is bounded above by the graph of y = x - $x ^ { 3 }$ and below by the x-axis on the interval from $x = 0$ to $x = 1$ Find the volume of the solid of revolution generated by revolving the region about the x-axis. Enter your answer as a reduced quotient of form $\frac { a b } { c }$ .

This is a sketch of the region between the two curves f(x) = $x ^ { 2 }$ , g(x) = 8 - $x ^ { 2 }$ . Compute this area. Enter just a reduced fraction of form $\frac { a } { b }$ .

Calculate. - $\int _ { 0 } ^ { 1 } \left( e ^ { 3 x } - \frac { 1 } { ( x + 1 ) ^ { 2 } } \right) d x$

Find all antiderivatives of the function. -f(x) = $x ^ { 5 }$ Enter your answer as a polynomial in x in standard form.

Suppose the the profit realized by a department store t days after its opening is given by the formula $4 t ^ { 3 } - 2 t + 1$ What was the average profit per day of the store during the first five days?

Suppose that a 1000-L water tank takes 20 min to drain and that after t minutes, the amount of water remaining in the tank is V(t) = $\frac { 5 } { 2 }$ (20 - $t ^ { 2 }$ ) liters. What is the average amount of water in the tank during the time it drains? Enter a reduced fraction of form $\frac { a } { b }$ (no units)

Use a Riemann sum to approximate the area under the graph of f(x) on the given interval. Use the right endpoints. f(x) = 3x - 4; 1 ≤ x ≤ 4, n = 6