Exam 5: Integrals

Express $\lim _ { n \rightarrow \infty } \left\{ \sum _ { i = 1 } ^ { n } \left[ \cos \left( \frac { \pi i } { n } \right) \right] \frac { \pi } { n } \right\}$ as a definite integral, then find the limit.

Determine whether each of the following statements is true or false. If the statement is true, provide an appropriate justification. If the statement is false, provide a counterexample.(a) If $f ( x ) \neq 0 \text { on } [ a , b ] \text { then } \int _ { a } ^ { b } f ( x ) d x \neq 0$ (b) If $f$ is continuous on an interval containing a,b, and c, then $\int _ { a } ^ { b } f ( x ) d x = \int _ { a } ^ { c } f ( x ) d x - \int _ { b } ^ { c } f ( x ) d x$ (c) If $\int _ { a } ^ { b } f ( x ) d x \geq 0 , \text { then } f ( x ) \geq 0 \text { on } [ a , b ]$

The velocity of a particle moving along a line is $t ^ { 3 } - t$ meters per second. Find the distance traveled in meters during the time interval $0 \leq t \leq 2$ .

Suppose using n = 10 to approximate the integral of a certain function by the Trapezoidal Rule results in an upper bound for the error equal to $\frac { 1 } { 10 }$ . What will the upper bound become if we change to n = 20?

Evaluate the improper integral $\int _ { 0 } ^ { \infty } x e ^ { - x / 2 } d x$

Evaluate the following integrals: (a) $\int _ { 0 } ^ { x / 4 } \sin 2 x \sin x d x$ (b) $\int e ^ { x } \sin \left( e ^ { x } \right) d x$ (c) $\int _ { - 1 } ^ { 2 } \frac { x } { \sqrt { x + 2 } } d x$ (d) $\int \tan x d x$

Let f (x) = $2 x + 1$ on the interval [ - 1,1]. Let the interval be divided into four equal subintervals. Find the value of the Riemann sum $\sum _ { i = 1 } ^ { 4 } f \left( x _ { i } ^ { * } \right) \Delta x _ { i } \text { if } x _ { i } ^ { * }$ is the midpoint of its subinterval.

Evaluate the following integrals: (a) $\int x ^ { 3 } \cos \left( x ^ { 4 } + 1 \right) d x$ (b) $\int \frac { \cos ( \sqrt { x } ) } { \sqrt { x } } d x$ (c) $\int \frac { \cos ( 1 / x ) } { x ^ { 2 } } d x$ (d) $\int \frac { \cos ( \arctan x ) } { 1 + x ^ { 2 } } d x$

Use the Comparison Theorem to determine whether $\int _ { 1 } ^ { \infty } \frac { \sin ^ { 4 } x } { x ^ { 2 } } d x$ is convergent or divergent. Justify your answer.

Evaluate the improper integral $\int _ { 0 } ^ { \infty } x ^ { - 1 / 2 } d x$

Let f (x) = $x ^ { 2 } - 2$ on the interval [0,2]. Let the interval be divided into four equal subintervals. Find the value of the Riemann sum $\sum _ { i = 1 } ^ { 4 } f \left( x _ { i } ^ { * } \right) \Delta x _ { i } \text { if } x _ { i } ^ { * }$ is the midpoint of the subinterval.

Let $f ( x ) = \int _ { 0 } ^ { x } \frac { \sin t } { t ^ { 2 } + 1 } d t . \text { Find } f ^ { \prime } ( x )$

Use the Trapezoidal Rule with n = 2 to approximate the integral $\int _ { 0 } ^ { 1 } x ^ { 3 } d x$

Use Simpson's Rule with n = 6 to approximate $\int _ { 0 } ^ { 2 } \sqrt { x ^ { 3 } + 2 } d x$

Find the value of $\int _ { e } ^ { e } \frac { d x } { x \sqrt { \ln x } }$

Evaluate the following integrals: (a) $\int x \sec ^ { 2 } x d x$ (b) $\int \left( 3 x ^ { 2 } + 1 \right) \ln \left( x ^ { 2 } + 1 \right) d x$ (c) $\int \frac { e ^ { \sqrt { x } } } { \sqrt { x } } d x$ (d) $\int e ^ { \sqrt { x } } d x$

Evaluate the following integrals: (a) $\int \frac { x } { x ^ { 2 } - x - 2 } d x$ (b) $\int \frac { 1 } { x ( x - 1 ) ^ { 2 } } d x$ (c) $\int \frac { 1 } { x ^ { 3 } + x } d x$ (d) $\int \frac { x ^ { 3 } + x + 2 } { x ^ { 2 } + 1 } d x$

What is the largest value of $\int _ { 0 } ^ { 3 } f ( x ) d x$ if $- 4 \leq f ( x ) \leq 2$ .

Sketch a plane region whose area is equal to $\lim _ { n \rightarrow \infty } \left\{ \sum _ { i = 1 } ^ { n } \left[ 4 - \frac { i } { n } \right] \frac { 1 } { n } \right\}$ then find the limit.

Use Simpson's Rule with n = 4 to approximate the integral $\int _ { - 2 } ^ { 2 } 2 ^ { x } d x$