Exam 6: The Definite Integral

Determine the average value of g(x) = $\sqrt { x }$ over the interval from x = 0 to x = 4. Enter a reduced fraction of form $\frac { a } { b }$ .

Find a function f(x) with the following property: f'(x) = - $\frac { 1 } { 4 - x }$ , f(2) = 0.

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

$\int _ { 2 } ^ { 5 } \left( e ^ { 4 x } - \frac { 1 } { x } \right) d x$ Enter your answer as a( $\mathrm { e } ^ { \mathrm { b } }$ - $\mathrm { e } ^ { \mathrm { C } }$ ) + d.

Find: $\int \left( x ^ { 3 } + 1 \right) d x$ Enter your answer as a polynomial in x in standard form with any fractional coefficients or powers reduced of form $\frac { a } { b }$ .

Suppose that at time t (0 ≤ t ≤ 2 , t in months), the sales of a certain commodity are decreasing at a rate of $1000 \mathrm { e } ^ { - 0.05t }$ units per month. Calculate the total change in sales from t = 0 to t = 2. Enter your answer as just a(1 - $\mathrm { e } ^ { \mathrm { b } }$ ) where b is a real number to one decimal place.

For the Riemann sum, [ $( 3.5 ) ^ { 4 }$ + 7(3.5) + $( 4 ) ^ { 4 }$ + 7(4) + $( 4.5 ) ^ { 4 }$ + 7(4.5) + $( 5 ) ^ { 4 }$ + 7(5)](0.5); a = 3, find n, b, and f(x). Enter your answer as just n, b, f(x) (2 integers in that order separated by commas and followed by a polynomial in x).

Find: $\int \left( x ^ { 2 } - \frac { 1 } { 4 x } \right) d x$ Enter your terms in standard forms in the order in which they appear in the integral.

Set up a definite integral or sum of definite integrals that represent the indicated shaded areas over the interval [a, c]. y = g(x)

What is the area under the curve y = $\frac { 2 } { x }$ between x = 1 and x = 3?

Find: $\int ( 2 x + 1 ) ^ { 2 } d x$ Enter your answer as a polynomial in x in standard form with any fractional coefficients or powers reduced of form $\frac { a } { b }$ .

Find a function f(x) with the following property: f'(x) = $x ^ { 6 }$ + $\frac { x ^ { 4 } } { 3 }$ - $x ^ { 3 }$ + $\frac { 5 } { 3 }$ $x ^ { 2 }$ , f(0) = e.

$\int _ { a } ^ { b } ( f ( x ) - g ( x ) ) d x$ expresses the area between the curves y = f(x) and y = g(x) from $x = a \operatorname { to } x = b$ only if:

Determine the average value of f(x) = $x ^ { 3 }$ + 3 $x ^ { 2 }$ + 3x + 1 over the interval from x = -1 to x = 3. Enter an integer.

Determine the average value of f(x) = $e ^ { x }$ over the interval from x = 1 to x = 4. Enter your answer in the form a( $\mathrm { e } ^ { \mathrm { b} }$ ± e)

Find the area bounded by y = $\frac { 1 } { x }$ , the x-axis, and the lines $x = 1 \text { and } x = a > 1$ Enter a real number (no approximations).

Find the value of k that makes the antidifferentiation formula true. - $\int x ^ { - 7 } d x$ = k $x ^ { - 6 }$ + C

Find all antiderivatives of the function. -f(x) = $x ^ { 15 }$ Enter your answer as a polynomial in x in standard form.

Determine the average value of f(x) = $\frac { 1 } { x ^ { 2 } }$ over the interval from x = 1 to x = 100. Enter a reduced fraction of form $\frac { a } { b }$ .

Find the area of the region bounded by f(x) = { $x ^ { 2 }$ (0 ≤ x ≤1); 2x - $x ^ { 2 }$ (1 ≤ x ≤ 2)} and the line $y = 0$ Enter just an integer.