Exam 5: Integrals

Use the Trapezoidal Rule with n = 4 to approximate the integral $\int _ { 1 } ^ { 3 } \frac { 1 } { x } d x$

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

Evaluate the following integrals: (a) $\int x ( 5 + \sqrt { x } ) d x$ (b) $\int _ { 0 } ^ { 1 } \frac { 5 } { 1 + x ^ { 2 } } d x$ (c) $\int _ { 0 } ^ { x / 6 } 1 + \cos \theta d \theta$ (d) $\int _ { x / 4 } ^ { x / 3 } \sec ^ { 2 } \theta d \theta$

Find the value of the integral $\int _ { 0 } ^ { x / 8 } \sin ^ { 2 } ( 4 x ) d x$

The work done against gravity in propelling an object with mass m kg to an altitude of h m above the surface of the earth is given by $W = \int _ { 637 \times 10 ^ { 6 } } ^ { 637 \times 10 ^ { 6 } + h } \frac { G M _ { E } m } { r ^ { 2 } } d r$ where 6.37 $\times$ 10⁶ m is the radius of the earth, $G \approx 6.667 \times 10 ^ { - 11 } \mathrm {~N} \cdot \mathrm { m } ^ { 2 } / \mathrm { kg } ^ { 2 }$ and $M _ { E } \approx 5.90 \times 10 ^ { 24 } \mathrm {~kg}$ is the mass of the earth.(a) Find the work required to launch a 1000-kilogram satellite vertically to an altitude of 1000 km.(b) The formula shows that the work is dependent on h. Show that $\int _ { 637 \times 10 ^ { 6 } } ^ { \infty } \frac { G M _ { Z } m } { r ^ { 2 } } d r$ is convergent. What is the physical significance of the fact that this improper integral is finite?

Let $f ( x ) = \int _ { - 3 } ^ { x } g ( t ) d t$ where g is the function whose graph is shown below: (a) Evaluate f ( $- 3$ ), f ( $- 1$ ), f (0), f (1), f (3) and f (5).(b) On what interval(s) is f increasing? (c) What are the maximum and minimum values of f over [ $- 3$ ,5]?

Evaluate the improper integral $\int _ { 0 } ^ { 3 x / 4 } \sec ^ { 2 } x d x$

Evaluate the following integrals: (a) $\int \frac { \ln x } { x ^ { 2 } } d x$ (b) $\int \frac { x ^ { 2 } + x + 1 } { x ^ { 2 } - 1 } d x$ (c) $\int \frac { \cos x } { 4 + \sin ^ { 2 } x } d x$ (d) $\int x ^ { 3 } \sqrt { 4 - x ^ { 2 } } d x$

Evaluate the improper integral $\int _ { 0 } ^ { 1 } \frac { \ln x } { x } d x$

Use the definition of the definite integral to evaluate $\int _ { 2 } ^ { 5 } \left( x ^ { 2 } - 3 x + 1 \right) d x$

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

If $\int _ { 0 } ^ { 3 } f ( x ) d x = 4 , \int _ { 3 } ^ { 6 } f ( x ) d x = 4 \text {, and } \int _ { 2 } ^ { 6 } f ( x ) d x = 5 \text {, find the value of } \int _ { 0 } ^ { 2 } f ( x ) d x \text {. }$

Sketch a plane region whose area is equal to $\lim _ { n \rightarrow \infty } \left\{ \sum _ { i = 1 } ^ { n } \left[ 2 + \left( \frac { i } { n } \right) ^ { 2 } \right] \frac { 1 } { n } \right\}$

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

The graph of a function f is given below. List the following quantities from smallest to largest. (a) $\int _ { 0 } ^ { 2 } f ( x ) d x$ (b) $\int _ { 0 } ^ { 2 } \sqrt { f ( x ) } d x$ (c) $\int _ { 0 } ^ { 2 } f ^ { 2 } ( x ) d x$ (d) $\int _ { 0 } ^ { 2 } f ^ { \prime } ( x ) d x$ (e) $\int _ { 0 } ^ { 1 } f ^ { Q } ( x ) d x$

Find the value of the integral $\int _ { 0 } ^ { x / 2 } \sin ( 2 t ) d t$

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

The graph of f below consists of line segments and semicircles. Use it to evaluate the following integrals. (a) $\int _ { 0 } ^ { 14 } f ( x ) d x$ (b) $\int _ { 0 } ^ { 10 } f ( x ) d x$ (c) $\int _ { 3 } ^ { 12 } f ( x ) d x$

(a) Estimate $\ln 2 = \int _ { 1 } ^ { 2 } \frac { 1 } { x } d x$ using Simpson's Rule with n = 4.(b) Estimate the error of the approximation in part (a).(c) How large should we take n to guarantee that the estimate by Simpson's Rule is accurate to within 0.001?

Use the Midpoint Rule with n = 6 to approximate $\int _ { 2 } ^ { 8 } \frac { 1 } { 1 + x ^ { 2 } } d x$