The boundaries of the shaded region are the y-axis, the line y = 1, and the curve y= V. Find the area of this region by writing a as a function of y and integrating with respect to y. YA 1 0 Area = y = 1 y = √√√x 1 X
The boundaries of the shaded region are the y-axis, the line y = 1, and the curve y= V. Find the area of this region by writing a as a function of y and integrating with respect to y. YA 1 0 Area = y = 1 y = √√√x 1 X
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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![**Problem Statement:**
The boundaries of the shaded region are the y-axis, the line \( y = 1 \), and the curve \( y = \sqrt[4]{x} \). Find the area of this region by writing \( x \) as a function of \( y \) and integrating with respect to \( y \).
**Diagram Explanation:**
- **Axes:** The diagram shows a standard Cartesian plane with the x- and y-axes.
- **Curve:** The red curve represents \( y = \sqrt[4]{x} \), which is a function that grows slowly and is concave up.
- **Line:** The horizontal line at \( y = 1 \) is a boundary for the shaded area.
- **Shaded Region:** The area of interest is shaded in yellow, bounded on the left by the y-axis (x = 0), above by the line \( y = 1 \), and to the right by the curve \( y = \sqrt[4]{x} \).
**Calculation:**
Area = \(\boxed{\phantom{0}}\)](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1a226d24-09f5-4831-a21f-7d913142a361%2F9cde1573-2ed4-42b8-ac16-d809114a3d74%2Fsbro0oe_processed.jpeg&w=3840&q=75)
Transcribed Image Text:**Problem Statement:**
The boundaries of the shaded region are the y-axis, the line \( y = 1 \), and the curve \( y = \sqrt[4]{x} \). Find the area of this region by writing \( x \) as a function of \( y \) and integrating with respect to \( y \).
**Diagram Explanation:**
- **Axes:** The diagram shows a standard Cartesian plane with the x- and y-axes.
- **Curve:** The red curve represents \( y = \sqrt[4]{x} \), which is a function that grows slowly and is concave up.
- **Line:** The horizontal line at \( y = 1 \) is a boundary for the shaded area.
- **Shaded Region:** The area of interest is shaded in yellow, bounded on the left by the y-axis (x = 0), above by the line \( y = 1 \), and to the right by the curve \( y = \sqrt[4]{x} \).
**Calculation:**
Area = \(\boxed{\phantom{0}}\)
![Consider the area between the graphs \( x + 3y = 13 \) and \( x + 5 = y^2 \). This area can be computed in two different ways using integrals.
First of all, it can be computed as a sum of two integrals
\[
\int_a^b f(x) \, dx + \int_b^c g(x) \, dx
\]
where \( a = \boxed{} \), \( b = \boxed{} \), \( c = \boxed{} \) and
\[
f(x) = \boxed{}
\]
\[
g(x) = \boxed{}
\]
Alternatively, this area can be computed as a single integral
\[
\int_{\alpha}^{\beta} h(y) \, dy
\]
where \( \alpha = \boxed{} \), \( \beta = \boxed{} \) and
\[
h(y) = \boxed{}
\]
Either way, we find that the area is \( \boxed{} \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F1a226d24-09f5-4831-a21f-7d913142a361%2F9cde1573-2ed4-42b8-ac16-d809114a3d74%2Fd566fhj_processed.jpeg&w=3840&q=75)
Transcribed Image Text:Consider the area between the graphs \( x + 3y = 13 \) and \( x + 5 = y^2 \). This area can be computed in two different ways using integrals.
First of all, it can be computed as a sum of two integrals
\[
\int_a^b f(x) \, dx + \int_b^c g(x) \, dx
\]
where \( a = \boxed{} \), \( b = \boxed{} \), \( c = \boxed{} \) and
\[
f(x) = \boxed{}
\]
\[
g(x) = \boxed{}
\]
Alternatively, this area can be computed as a single integral
\[
\int_{\alpha}^{\beta} h(y) \, dy
\]
where \( \alpha = \boxed{} \), \( \beta = \boxed{} \) and
\[
h(y) = \boxed{}
\]
Either way, we find that the area is \( \boxed{} \).
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