Exam 5: Integrals

Given the graph below, use 5 rectangles to estimate the area under the graph from x = 0 to x = 5. Compute L₅ (sample points are left endpoints), R₅ (sample points are right endpoints) and M₅ (sample points are midpoints). Which of the estimates appears to give the best estimate? Justify your answer.

Use Definition 2 to show that the area under the curve $y = 2 ^ { x } , 0 \leq x \leq 1 \text { is } \frac { 1 } { \ln 2 }$

Evaluate the following integrals: (a) $\int \frac { 2 } { x ^ { 2 } } + \frac { 1 } { x } d x$ (b) $\int e ^ { x } + \frac { 5 } { \sqrt [ 3 ] { x ^ { 2 } } } d x$ (c) $\int \sec \theta \cdot \tan \theta d \theta$ (d) $\int 5 ^ { 2 } d t$

Evaluate the following integrals: (a) $\int x \sin x d x$ (b) $\int x \sin \left( x ^ { 2 } \right) d x$ (c) $\int \sin ^ { - 1 } x d x$ (d) $\int x e ^ { 3 x } d x$

Two students use Simpson's Rule to estimate $\int _ { 1 } ^ { 2 } \left( x ^ { 3 } + 2 x ^ { 2 } - x + 1 \right) d x$ . One divides the interval into 30 equal subintervals and the other into 60 equal subintervals. How will the accuracy of their estimates compare?

Let $f ( x ) = \int _ { 3 x } ^ { 2 } \frac { \sin t } { t ^ { 2 } + 1 } d t . \text { Find } f ^ { \prime } ( x )$ A) $\frac { \sin 3 x } { 3 x ^ { 2 } + 1 }$ B) $- \frac { 3 \sin 3 x } { 3 x ^ { 2 } + 1 }$ C) $\frac { \cos x } { x ^ { 2 } + 1 }$ D) $- \frac { 3 \sin 3 x } { 9 x ^ { 2 } + 1 }$ E) $\frac { 3 \sin 3 x } { 9 x ^ { 2 } + 1 }$ F) $- \frac { \cos 3 x } { 9 x ^ { 2 } + 1 }$ G) $\frac { \cos x } { 2 x }$ H) $- \frac { \cos x } { 2 x }$

Let $f ( x ) = \int _ { x } ^ { 10 } t ^ { 3 } d t \text {. Find the value of } f ^ { \prime } ( 2 ) \text {. }$

(a) Find $g ( t )$ so that $\int g ( t ) d t = \frac { t } { 2 } \sqrt { 4 - t ^ { 2 } } + 2 \cdot \sin ^ { - 1 } \frac { t } { 2 }$ (b) Evaluate $\int _ { 0 } ^ { 2 } g ( t ) d t$ using the Evaluation Theorem.(c) Evaluate $\int _ { 0 } ^ { 2 } g ( t ) d t$ by the area interpretation of the integral. Compare your answer with part (b) above.(d) Evaluate $\int _ { 0 } ^ { \sqrt { 3 } } g ( t ) d t .$

Find the value of $\int _ { 0 } ^ { \operatorname { In3 } } e ^ { 4 x } d x$

Suppose f (x) is a decreasing function on the interval $1 \leq x \leq 4$ . If you estimate $\int _ { 1 } ^ { 4 } f ( x ) d x$ using right endpoints, explain why the estimate is an overestimate or an underestimate.

If $\int _ { 0 } ^ { x ^ { 2 } } f ( t ) d t = x \ln x$ , where f is a continuous function, find f (1).

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

Use the definition of the definite integral to evaluate $\int _ { 0 } ^ { 2 } ( 1 - 2 x ) d x$

Let $f ( x ) = \frac { 1 } { 2 } \int _ { 0 } ^ { x } t ^ { 3 } d t \text {. Find the value of } f ^ { \prime } ( 2 ) \text {. }$

Two students use Simpson's Rule to estimate $\int _ { 0 } ^ { 1 } \left( x ^ { 2 } + 4 x - 5 \right) d x$ . One divides the interval into 30 equal subintervals and the other into 60 equal subintervals. How will the accuracy of their estimates compare?

Find the value of $\int _ { e } ^ { e ^ { 2 } } \frac { ( \ln x ) ^ { 2 } } { x } d x$

Find the partial fraction expansion of the rational function: $\frac { x ^ { 2 } } { ( x - 1 ) \left( x ^ { 2 } + 1 \right) }$ .

Let $f ( x ) = \int _ { 0 } ^ { x ^ { 2 } } \sqrt [ 3 ] { 4 - t ^ { 3 } } d t . \text { Find } f ^ { \prime } ( x )$

Evaluate the improper integral $\int _ { 1 } ^ { \infty } \frac { 1 } { x ( x + 1 ) } d x$

Find the value of the integral $\int _ { 1 } ^ { 4 } \frac { \ln x } { x ^ { 2 } } d x$