Exam 6: The Definite Integral

Calculate. - $\int _ { 0 } ^ { 1 } \frac { 2 } { 3 - 2 x } d x$

Find: $\int \left( 3 x ^ { 1 / 3 } \right) d x$ . Enter using standard power function form a $x ^ { b }$ , with any fractions reduced of form $\frac { a } { b }$ .

Use a Riemann sum to approximate the area under the graph of $f ( x ) = x , 0 \leq x \leq 5 , n = 10$ Use the right endpoints. Enter just a real number to two decimal places.

Determine if the function F is the general antiderivative of the function f. F(x) = 5x + C; f(x) = 5

Find the area under the curve y = $\frac { 1 } { x }$ - 2x from x = -3 to x = -2. Enter a ± ln b using reduced fractions of form $\frac { a } { b }$ and integers.

Determine the average value of f(x) = $x ^ { 3 }$ over the interval from x = 0 to x = 4. Enter an integer.

What is the area under the curve y = $\frac { 1 } { \sqrt { x } }$ between x = 1 and x = 2?

Find: $\int 2 \mathrm { e } ^ { - 2 x }$ dx Enter your answer in standard form (no fractions).

$\int _ { 1 } ^ { 4 } \sqrt { x } d x$ Enter just a reduced fraction of form $\frac { a } { b }$ .

Suppose the $2500 is deposited in a savings account paying 5% interest, compounded continuously. What will be the average value of the account during the next 10 years?

Draw the region whose area is given by the definite integral. $\int _ { 1 } ^ { 3 } x ^ { 2 }$ dx

This is a sketch of the region between the two curves f(x) = $x ^ { 3 }$ , g(x) = $x ^ { 4 }$ . Compute this area. Enter your answer as just a reduced fraction of form $\frac { a } { b }$ .

Find the area of the region bounded by the curves y = $x ^ { 2 }$ + $\frac { 8 } { 5 }$ x + 1 and y = $\frac { 2 } { 5 }$ x + 1. Enter just a reduced fraction of form $\frac { a } { b }$ .

Given $f ( x ) = \ln ( x + 1 ) ; \quad 0 \leq x \leq 1 , n = 3 ,$ set up a Riemann sum to approximate the area under the graph of f(x) on the given interval. Use the midpoints. Is the following the correct answer? $\ln \left( \frac { 7 } { 6 } \right) + \ln \left( \frac { 3 } { 2 } \right) + \ln \left( \frac { 11 } { 6 } \right)$

Find: $\int \left( \sqrt { x } + \frac { 1 } { \sqrt { x } } \right) d x$ Enter your answer using standard power function form (a $x ^ { b }$ ), leaving the terms in the order in which they appear in the integral.

$\int _ { - 100,000 } ^ { 100,000 } x ^ { 3 } d x$ Enter just an integer.

Find a function f(x) with the following property: f'(x) = 2 $\mathrm { e } ^ { 2 x }$ - $\frac {4} { \mathrm { x }}$ + $\frac { 3 } { x ^ { 2 } }$ , f(1) = -2.

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

A newspaper is launching a new advertising campaign in order to increase the number of daily subscribers. The newspaper currently (t = 0) has 26,000 daily subscribers and management expects that number, S(t), to grow at the rate of $S ^ { \prime } ( t ) = 80 t ^ { 1 / 2 }$ subscribers per day, where t is the number of days since the campaign began. How long (to the nearest day) should the campaign last if the newspaper wants the number of daily subscribers to grow to 49,000?

Refer to the information in the graph below. Given functions f(x) = $x ^ { 2 }$ + 4x + 4 and g(x) = $x ^ { 2 }$ - 4x + 4, set up a definite integral or sum of definite integrals that gives the area of the shaded portion.