4.1.3How do the limits of integration work?

More Properties of Definite Integrals

4-27.

PROPERTIES OF DEFINITE INTEGRALS

Consider the integral expressions below. For each expression, draw and shade the region for a generic function. Simplify each integral expression and summarize each case on your paper.

$\int _ { a } ^ { a } f ( x ) d x$
$\int _ { a } ^ { b } f ( x ) d x + \int _ { b } ^ { c } f ( x ) d x$
$\int _ { b } ^ { a } f ( x ) d x + \int _ { a } ^ { b } f ( x ) d x$
$\int _ { b } ^ { a } k \cdot f ( x ) d x$ , where $k$ is a constant

4-28.

PROPERTIES OF DEFINITE INTEGRALS, CONTINUED

You have developed methods of simplifying integral expressions with a single function. What happens when we combine definite integrals with two different functions? Investigate the following relationship:
$\int _ { a } ^ { b } f ( x ) d x + \int _ { a } ^ { b } g ( x ) d x$

Evaluate $\int _ { 0 } ^ { 2 } x d x + \int _ { 0 } ^ { 2 } 3 x d x$ .
Evaluate $\int _ { 0 } ^ { 2 } ( 4 x ) d x$ .
Rewrite the expression $\int _ { a } ^ { b } f ( x ) d x + \int _ { a } ^ { b } g ( x ) d x$ into a simplified form.

4-29.

TRANSLATIONS OF FUNCTIONS

Examine what happens to the area of a region when a function is translated. Some cases to consider are listed below, but do not feel restricted to them. When finished, summarize your findings clearly.

Does $\int _ { a } ^ { b } [ f ( x ) + k ] d x = \int _ { a } ^ { b } f ( x ) d x + k$ ? Explain why or why not.
Does $\int _ { a } ^ { b } f ( x ) d x = \int _ { a + c } ^ { b + c } f ( x ) d x$ ? Explain why or why not.
Does $\int _ { a } ^ { b } f ( x ) d x = \int _ { a + c } ^ { b + c } f ( x - c ) d x$ ? Explain why or why not.
Does $\int _ { a } ^ { b } f ( x ) d x = \int _ { a } ^ { b } f ( x + c ) d x$ ? Explain why or why not.
Summarize the definite integral properties that are correct on your paper.

Continuous curve labeled, f of x, coming from lower left, turning down half way up on y axis, then up in first quadrant, with 2 unequal tick marks on the x axis, labeled, a, & b, with region between the curve & x axis, & between x = A, & x = b, shaded.

4-30.

With your team, write general formulas for all the properties of definite integrals you discovered today.

4-31.

Differentiate the following equations with respect to $x$ . That is, what is $\frac { d y } { d x }$ ? Homework Help ✎

$y = \frac { x + 1 } { x }$
$y=\cos\left(x\right)+\sin\left(x\right)$
$y = x \cdot \sqrt [ 3 ] { x ^ { 2 } }$
$y = \left(6 – 5x\right)\left(1 – 2x\right)$

4-32.

Evaluate the following definite integrals without a calculator. Then write a statement about the connection between them. Check your answers with a calculator. Homework Help ✎

Compute without a calculator

$\int _ { 2 } ^ { 9 } 8 x d x$
$\int _ { 2 } ^ { 9 } ( 8 x + 5 ) d x$
$\int _ { 2 } ^ { 9 } 5 d x$

4-33.

Given the graph at right of $f\left(x\right) = 2x + 1$ , evaluate: Homework Help ✎

$\int _ { 1 } ^ { 3 } ( 2 x + 1 ) d x$
$\int _ { 1 } ^ { 3 } ( 2 t + 1 ) d t$
What is the difference between the expressions in parts (a) and (b)?

Increasing line labeled, f of x = 2 x + 1, passing through the points (0, comma 1), & (3, comma 7), with region between the line & the x axis, & between x = 1, & x = 3, shaded.

4-34.

Write the equations of the two lines tangent to the curve $f\left(x\right) = x^{3} – x^{2} + x + 1$ that have a slope of $2$ . 4-34 HW eTool Homework Help ✎
Determine the equations of the lines perpendicular to the tangent lines from part (a) at their points of tangency to $f$ .

4-35.

Given $f\left(x\right)=\sin\left(x\right),g\left(x\right)=x^2$ and $h(x)=\frac{1}{x}$ , use compositions of functions to express each of the following functions. Homework Help ✎

$y=\sin\left(x^2\right)$
$y=\sin^2\left(x\right)$
$y=\csc\left(x\right)$
$y = \operatorname { csc } ^ { 2 } ( \frac { 1 } { x } )$

4-36.

Using the distance vs. time graph at right, determine if the velocity is positive, negative, or zero at each labeled point on the graph. Homework Help ✎

First quadrant, continuous curve coming from upper left, passing through highlighted point labeled, a, turning up at highlighted point labeled b, changing from concave up to concave down at highlighted point c, turning down at highlighted point d, passing through highlighted point e, stopping at the x axis.

4-37.

Sketch a graph of $f\left(x\right) = x^{3} – 2x^{2}$ . At what point(s) will the line tangent to $f$ be parallel to the secant line through $\left(0,f\left(0\right)\right)$ and $\left(2,f\left(2\right)\right)$ ? 4-37 HW eTool Homework Help ✎

4-38.

Sketch a graph of $f\left(x\right) = x^{3} + 3x^{2} – 45x + 8$ . Homework Help ✎

Calculate the slope of the line tangent to the curve at $x = –2$ .
Determine the point on the curve where the slope is the smallest (steepest negative slope). What is the name of this point?

4-39.

Let $f ( x ) = \left\{ \begin{array} { l l } { 2 x ^ { 2 } - 4 \text { for } x \leq 3 } \\ { - 2 x - 5 \text { for } x > 3 } \end{array} \right.$ . Homework Help ✎

What is $\lim\limits _ { x \rightarrow 3 ^ { + } } f ( x )$ ?
What is $\lim\limits _ { x \rightarrow 3 ^ { - } } f ( x )$ ?
What do your results from parts (a) and (b) tell you about $f$ ?

Calculus Third Edition