Exam 6: The Definite Integral

A rock is dropped from a balloon hovering at 4800 ft above the ground. Its velocity at time t seconds is $v ( t ) = - 32 t$ feet per second. Find how long it takes for the rock to reach the ground. Enter just a real number to one decimal place (no units).

Find: $\int \frac { 1 + x ^ { 2 } } { x ^ { 2 } } d x$ Enter terms in the order in which they appear in standard power function forms with any constant at the right end.

This is a sketch of the region between the two curves f(x) = $x ^ { 2 }$ , g(x) = $x ^ { 3 }$ . Compute the area. Enter your answer as just a reduced fraction of form $\frac { a } { b }$ .

Find: $\int - 3 \left( e ^ { x / 2 } + 1 \right) d x$ Enter your terms in the same order in which they appear in the integral with each term in standard power function form; any fractions reduced form $\frac { a } { b }$ .

Determine the average value of f(x) = x - $x ^ { 2 }$ over the interval from x = 0 to x = 1. Enter a reduced fraction of form $\frac { a } { b }$ .

Find all antiderivatives of the function. -f(x) = $x ^ { 3 }$ Enter your answer as a polynomial in x in standard form.

Given f(x) = $x ^ { 3 }$ - 1 on the interval 1 ≤ x ≤ 5 and with n = 4, compute the Riemann sum (a) using the left endpoints; (b) using the right endpoints; and (c) using the midpoints of the subintervals. Enter your answer as just a, b, c all integers separated by commas. Enter the numbers in the order that answers (a), (b), (c) but do not label.

If A(t) denotes the annual rate of world consumption of oil at time t (with t = 0 corresponding to 1977), which of the following expressions represents the amount of oil consumed between 1977 and 1987?

Find all functions f(x) with the following property: f'(x) = $x ^ { 5 }$ + 2 $x ^ { 3 }$ - $3 x ^ { 2 }$ + 6. Enter your answer as a polynomial in x in standard form with any fractional coefficients or powers reduced of form $\frac { a } { b }$ .

Determine Δx formed by partitioning the given interval into n subintervals. -8 ≤ x ≤ -4; n = 6

A certain commodity has demand curve p = $\frac { 20 } { x + 5 }$ - 1 at sales level x. What is the consumers' surplus if 5 units are currently being sold?

$\int _ { 1 } ^ { 3 } \left( x ^ { 3 } + 3 \right) x ^ { 2 } d x$ Enter a reduced fraction of form $\frac { a } { b }$ .

Find the area under the curve y = $\frac { 1 } { e ^ { x } }$ + $\frac { 1 } { x - 2 }$ between x = 3 and x = 5. Enter your answer in the form $e ^ { a }$ - $\mathrm { e } ^ { \mathrm { b } }$ + c (no approximations).

This is a sketch of the region between the two curves y = $x ^ { 2 }$ - 15, below the x-axis and also below the line $y = - 2 x$ Does the following represent the area of the region? $\int _ { - \sqrt { 15 } } ^ { 0 } \left( x ^ { 2 } - 15 \right) d x + \int _ { 0 } ^ { 3 } \left( - x ^ { 2 } - 2 x + 15 \right) d x$

$\int _ { 2 } ^ { 10 } \frac { 1 } { x - 1 }$ dx Enter just a real number (no approximations).

Find the area of the region bounded by y = $e ^ { - 2 x }$ , the x-axis, and the lines $x = 1 \text { and } x = 3 \text {. }$ Enter your answer in the form a( $\mathrm { e } ^ { \mathrm { b } }$ - $e ^ { c })$ .

What is the consumers' surplus for the demand curve p = 5 - $\frac { x } { 20 }$ at the sales level $x = 60$ ?

Given f(x) = lnx on the interval 1 ≤ x ≤ 5 and with n = 2, compute the Riemann sum (a) using the left endpoints; (b) using the right endpoints; and (c) using the midpoints of the subintervals. Enter your answer as just a, b, c all real numbers rounded to two decimal places and separated by commas. Enter the numbers in the order that answers (a), (b), (c) but do not label.

$\int _ { 3 } ^ { 5 } e ^ { 5 x } d x$ Enter your answer as a( $\mathrm { e } ^ { \mathrm { b} }$ - $\mathrm { e } ^ { \mathrm { C } }$ ).

Determine the area under the curve y = $\frac { 1 } { x }$ from x = 1 to x = e. Enter just an integer.