4.1.2How accurate is a Riemann sum?

Use the numerical integration feature of a graphing calculator to evaluate each of the integrals below. Then, for the region associated with each integral, evaluate a Riemann sum to approximate the area under the curve with $20$ rectangles. Finally, compare these results.

1. $\int _ { - \pi } ^ { \pi / 2 } 3 \operatorname { sin } ( 2 x ) d x$ 

2. $\int _ { - 5 } ^ { 5 } ( x ^ { 2 } - 3 ) d x$ 

Draw and shade the region representing $\int _ { 5 } ^ { 0 } 4 x d x$.

1. Evaluate the integral geometrically and then verify your result using the numerical integration feature of a graphing calculator. What happened? 

2. Why is $\int _ { 5 } ^ { 0 } 4 x d x$ different than $\int _ { 0 } ^ { 5 } 4 x d x$? 

Does $\int _ { a } ^ { b } f ( x ) d x = \int _ { b } ^ { a } f ( x ) d x$ if $a ≠ b$? Why or why not?
Examine the limit of a Riemann sum, $\lim\limits _ { n \rightarrow \infty } \displaystyle\sum _ { i = 0 } ^ { n - 1 } \frac { b - a } { n } \cdot f ( a + \frac { b - a } { n } \cdot i )$ as you answer this question. 

COMBINING REGIONS

Rewrite each of the following expressions as a single integral.

1. $\int _ { 1 } ^ { 6 } f ( x ) d x - \int _ { 3 } ^ { 6 } f ( x ) d x$ 

2. $\int _ { 3 } ^ { 10 } f ( x ) d x + \int _ { 9 } ^ { 3 } f ( x ) d x$ 

3. $\int _ { c } ^ { d } f ( x ) d x + \int _ { e } ^ { c } f ( x ) d x$ 

4. $\int _ { a } ^ { x + h } f ( t ) d t - \int _ { a } ^ { x } f ( t ) d t$ 

5. When can we combine multiple regions? When can we rewrite them? Use the above examples to justify your answer. 

Evaluate the following definite integrals. Homework Help ✎

1. $- \int _ { 0 } ^ { 3 } x d x$ 

2. $\int _ { 0 } ^ { 3 } ( - x ) d x$ 

3. $\int _ { 3 } ^ { 0 } x d x$ 

4. $- \int _ { 3 } ^ { 0 } ( - x ) d x$ 

For each function below, write the equation of its general antiderivative, $F$. Homework Help ✎

1. $f\left(x\right) = –2$ 

2. $f\left(x\right)=\frac{3}{2}x^{−1/2}$ 

3. $f\left(x\right) = –3x^{2} + 6x$ 

4. $f\left(x\right) = 2\left(x + 3\right)$ 

Differentiate each function below. That is, write an equation for the slope function, $f^\prime$. Homework Help ✎

1. $f\left(x\right) = 6\left(x – 2\right)^{3}$ 

2. $f\left(x\right)=2\sin\left(x\right)$ 

3. $f\left(x\right) = \left(x + 5\right)\left(2x – 1\right)$ 

4. $f ( x ) = \large\frac { x ^ { 3 } - 6 x ^ { 2 } + 2 x } { x }$ 

Describe what a “slope function” is in complete sentences. What is its purpose? Give some examples of functions and their slope functions. Homework Help ✎

The value of $\int _ { 0 } ^ { \pi } x \cdot \operatorname { sin } ( x ) d x$ is $π$ . 4-22 HW eTool Homework Help ✎

1. Without a calculator, evaluate  $2 \int _ { 0 } ^ { \pi } x \cdot \operatorname { sin } ( x ) d x$ and $\int _ { - \pi } ^ { \pi } x \cdot \operatorname { sin } ( x ) d x$.

2. Use a graph to justify your conclusion.

For parts (a) and (b) below, trace $f\left(x\right)$ on your paper. Then, using a different color, sketch the graph of $y = f^\prime\left(x\right)$ for the function given. Homework Help ✎

Evaluate the following limits. Homework Help ✎

1. $\lim \limits _ { x \rightarrow 2 } \frac { x ^ { 2 } - 4 } { x - 2 }$ 

2. $\lim \limits _ { h \rightarrow 0 } \frac { ( 1 + h ) ^ { 2 } - 1 } { h }$ 

A cone has a height of $10$ inches and a radius of $4$ inches. If a plane cuts the cone $h$ inches above the base of the cone, write an expression for the area of the circular cross-section. Homework Help ✎