b Explain why h(x)dx does not represent the area between the graph of y=h(x) and the x •Sh(x)dx a axis from x = a to x = b in the figure to the right. Choose the correct answer below. 0 O A. The top boundary of the shaded region is y = 0, so h(x)dx represents the shaded area. Sh(x)dx a b B. The shaded region is below the x-axis, so ſh(-x)dx represents the shaded area. j a OC. The top boundary of the shaded region is y = 0, so 0 b b a h(x)dx represents the shaded area. O D. The shaded region is below the x-axis, so h(x)dx represents the negative of the area. h(x) a b y=h(x)
b Explain why h(x)dx does not represent the area between the graph of y=h(x) and the x •Sh(x)dx a axis from x = a to x = b in the figure to the right. Choose the correct answer below. 0 O A. The top boundary of the shaded region is y = 0, so h(x)dx represents the shaded area. Sh(x)dx a b B. The shaded region is below the x-axis, so ſh(-x)dx represents the shaded area. j a OC. The top boundary of the shaded region is y = 0, so 0 b b a h(x)dx represents the shaded area. O D. The shaded region is below the x-axis, so h(x)dx represents the negative of the area. h(x) a b y=h(x)
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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![### Understanding Definite Integrals and Area Representation
#### Problem Statement:
Explain why \(\int_a^b h(x)dx\) does not represent the area between the graph of \(y = h(x)\) and the x-axis from \(x = a\) to \(x = b\) in the figure to the right.
#### Graph Explanation:
There is a graph to the right showing the function \(y = h(x)\) which is entirely below the x-axis between the points \(x = a\) and \(x = b\).
#### Explanation:
When a function \(y = h(x)\) is below the x-axis, the definite integral \(\int_a^b h(x)dx\) computes the signed area, which will be negative. To find the actual area between the curve and the x-axis, we need to consider the absolute value of the integrand.
#### Multiple_choice Options:
Choose the correct answer below.
**A.** The top boundary of the shaded region is \(y = 0\), so \(\int_a^0 h(x)dx\) represents the shaded area.
\[ \int_a^0 h(x)dx \]
**B.** The shaded region is below the x-axis, so \(\int_a^b h(-x)dx\) represents the shaded area.
\[ \int_a^b h(-x)dx \]
**C.** The top boundary of the shaded region is \(y = 0\), so \(\int_0^b h(x)dx\) represents the shaded area.
\[ \int_0^b h(x)dx \]
**D.** The shaded region is below the x-axis, so \(\int_a^b h(x)dx\) represents the **negative** of the area.
\[ \int_a^b h(x)dx \]
*Correct answer: **D***](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F2b1cab43-2155-41da-a401-15f3493c67ce%2F45264503-b98b-470e-8ba9-6be67d0b3123%2F67tq61_processed.png&w=3840&q=75)
Transcribed Image Text:### Understanding Definite Integrals and Area Representation
#### Problem Statement:
Explain why \(\int_a^b h(x)dx\) does not represent the area between the graph of \(y = h(x)\) and the x-axis from \(x = a\) to \(x = b\) in the figure to the right.
#### Graph Explanation:
There is a graph to the right showing the function \(y = h(x)\) which is entirely below the x-axis between the points \(x = a\) and \(x = b\).
#### Explanation:
When a function \(y = h(x)\) is below the x-axis, the definite integral \(\int_a^b h(x)dx\) computes the signed area, which will be negative. To find the actual area between the curve and the x-axis, we need to consider the absolute value of the integrand.
#### Multiple_choice Options:
Choose the correct answer below.
**A.** The top boundary of the shaded region is \(y = 0\), so \(\int_a^0 h(x)dx\) represents the shaded area.
\[ \int_a^0 h(x)dx \]
**B.** The shaded region is below the x-axis, so \(\int_a^b h(-x)dx\) represents the shaded area.
\[ \int_a^b h(-x)dx \]
**C.** The top boundary of the shaded region is \(y = 0\), so \(\int_0^b h(x)dx\) represents the shaded area.
\[ \int_0^b h(x)dx \]
**D.** The shaded region is below the x-axis, so \(\int_a^b h(x)dx\) represents the **negative** of the area.
\[ \int_a^b h(x)dx \]
*Correct answer: **D***
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