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Chapter 4: Q. 5 (page 384)

Suppose f is positive on (鈭掆垶, 鈭�1] and [2,鈭�) and negative on the interval [鈭�1, 2]. Write (a) the signed area and (b) the absolute area between the graph of f and the x-axis on [鈭�3, 4] in terms of definite integrals that do not involve absolute values.

Short Answer

Expert verified

(a). ${鈭�}_{- 3}^{4} f (x) d x$

(b). ${鈭�}_{鈭� 3}^{鈭� 1} 鈥� f (x) d x 鈭� {鈭�}_{鈭� 1}^{2} 鈥� f (x) d x + {鈭�}_{2}^{4} 鈥� f (x) d x$

Step by step solution

01

(a). Given Information

$A function f is positive on (鈭� 鈭�, 鈭� 1]; [2, 鈭�) and negative on [鈭� 1, 2]$

$The objective is to write the signed area on [鈭� 3, 4] .$

$The signed area will be, {鈭�}_{- 3}^{4} f (x) d x$

$Therefore, the signed area is {鈭�}_{鈭� 3}^{4} 鈥� f (x) d x .$

02

(b). Given Information

$The objective is to write the absolute area on [鈭� 3, 4] without using the absolute area absolute$ values.

$The intervals will be [鈭� 3, 鈭� 1], [鈭� 1, 2], [2, 4]$

$The absolute area will be$

${鈭�}_{鈭� 3}^{鈭� 1} 鈥� f (x) d x 鈭� {鈭�}_{鈭� 1}^{2} 鈥� f (x) d x + {鈭�}_{2}^{4} 鈥� f (x) d x$

$Therefore, the signed area is {鈭�}_{鈭� 3}^{鈭� 1} 鈥� f (x) d x 鈭� {鈭�}_{鈭� 1}^{2} 鈥� f (x) d x + {鈭�}_{2}^{4} 鈥� f (x) d x .$

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Most popular questions from this chapter

Use the Fundamental Theorem of Calculus to find the exact values of the given definite integrals. Use a graph to check your answer.

${鈭�}_{蟺 / 4}^{蟺 / 2} 鈥� \csc 鈦� x \cot 鈦� x d x$

Given a simple proof that ${鈭�}_{k = 5}^{n} (a_{k} + b_{k}) = {鈭�}_{k = 5}^{n} a_{k} + {鈭�}_{k = 5}^{n} b_{k}$

Show that $F (x) = \frac{1}{2} e^{(x^{2})} (x^{2} - 1)$ is an anti-derivative of $f (x) = x^{3} e^{(x^{2})} .$

Approximations and limits: Describe in your own words how the slope of a tangent line can be approximated by the slope of a nearby secant line. Then describe how the derivative of a function at a point is defined as a limit of slopes of secant lines. What is the approximation/limit situation described in this section?

Describe an example that illustrates that ${鈭�}_{a}^{b} | f (x) | d x$ is not equal to $| {鈭�}_{a}^{b} f (x) d x |$ .

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