Exam 5: Integrals

Two cars start from rest at a traffic light. Car 1 travels at a speed given by $v _ { 1 } ( t ) = \frac { 1 } { 2 } t$ feet per minute, and Car 2 travels at a speed given by $v _ { 2 } ( t ) = \frac { 1 } { 4 } t ^ { 2 },$ feet per minute.(a) Which car is ahead after 2 minutes? Justify.(b) Which car is ahead after 3 minutes? Justify.(c) Which car is ahead after 4 minutes? Justify.(d) At what time(s) do the two cars travel at the same speed?

Use the Trapezoidal Rule with n = 1 to approximate the integral $\int _ { 0 } ^ { 1 } ( \sqrt { x } + 2 ) d x$

If $F ( x ) = \int _ { 2 } ^ { 3 x } e ^ { t ^ { 4 } } d t \text {, find the value of } F ^ { \prime } ( 0 ) \text {. }$

Find the value of the integral $\int _ { 4 } ^ { 6 } 2 x d x$

Compute $\frac { d } { d x } \int _ { x } ^ { 4 } \sin \left( t ^ { 3 } \right) d t$

Find the value of the integral $\int _ { 0 } ^ { x / 4 } \sin ^ { 2 } x \cos x d x$

Find the value of the integral $\int _ { 0 } ^ { x } x \cos x d x$

Evaluate the improper integral $\int _ { 1 } ^ { 2 } \frac { 1 } { \sqrt { 4 - x ^ { 2 } } } d x$

Find the value of the integral $\int _ { 1 } ^ { e } \ln x d x$

(a) Use integration by parts to prove the reduction formula: $\int x ^ { n } \cos x d x = x ^ { n } \sin x - n \int x ^ { n - 1 } \sin x d x$ (b) Demonstrate your understanding of this formula by using it to evaluate: $\int x ^ { 4 } \cos x d x$

Find the value of the integral $\int _ { 1 } ^ { 2 } \frac { x ^ { 2 } - 1 } { x } d x$

Let $f ( x ) = \int _ { x } ^ { 2 } \frac { \sin t } { t ^ { 2 } + 1 } d t . \text { Find } f ^ { \prime } ( x )$

Use the Table of Integrals in your textbook to evaluate each of the following: (a) $\int e ^ { x } \sec e ^ { x } d x$ (b) $\int \sin ^ { 4 } x \cos ^ { 2 } x d x$ (c) $\int \frac { 1 } { \cos ^ { 3 } x } d x$ (d) $\int x ^ { 3 } \sin x ^ { 2 } d x$ (e) $\int \frac { 1 } { \sqrt { 4 - 9 x ^ { 2 } } } d x$ (f) $\int _ { 0 } ^ { 2 } \frac { x ^ { 2 } } { \sqrt { 16 - x ^ { 2 } } } d x$ (g) $\int _ { 2 } ^ { 4 } \frac { d x } { \sqrt { 6 x - x ^ { 2 } - 5 } }$ (h) $\int _ { 0 } ^ { \pi / 2 } \frac { \sin t } { 1 + \cos ^ { 2 } t } d t$ (i) $\int _ { 2 } ^ { 5 } \frac { \sqrt { 4 + x ^ { 2 } } } { x ^ { 4 } } d x$ (j) $\int _ { 0 } ^ { x / 2 } e ^ { - x } \sin x d x$

Use left endpoints with n = 4 to approximate $\int _ { 1 } ^ { 3 } \ln x d x$

Find the value of the integral $\int _ { 0 } ^ { 1 } x e ^ { x } d x .$

Given the graph below, use 3 rectangles to find an estimate for the area under the graph from x = 0 to x = 9. Find lower and upper estimates for the area.

Estimate $\int _ { 2 } ^ { 5 } 1 + x ^ { 2 } d x$ using three approximating rectangles and taking the sample points to be left endpoints. Use a picture to explain why your estimation is an overestimate or an underestimate.

The velocity of a particle moving along a line is 1 - 2t meters per second. Find the distance traveled in meters during the time interval $0 \leq t \leq 1$ .

Evaluate the integral $\int \frac { 1 + \ln x } { x \ln x } d x$

Determine whether each integral is convergent or divergent. Evaluate those that are convergent.(a) $\int _ { 3 } ^ { \infty } \frac { 1 } { x ^ { 3 / 2 } } d x$ (b) $\int _ { e } ^ { \infty } \frac { 1 } { x ( \ln x ) } d x$ (c) $\int _ { 0 } ^ { \infty } \frac { x } { \left( x ^ { 2 } + 5 \right) ^ { 2 } } d x$