Suppose the function y =h(x) is nonnegative and continuous on [x.³], which implies that the area bounded by the graph of h and the x-axis on [a.B] equals h(x) dx or ay dx. If the graph of y = h(x) on [a.B) is traced exactly once by the parametric equations x = f(t), y = g(t), for a ≤t≤ b, then it follows by substitution that the area bounded by his given by the equation below. ["h(x) dx = [y dx = [g(t) f' (t) dt, if œ=f(a) and ß=f(b) (or √n(x) dx =) ) dx =g(t) f'(t) dt, if x = f(b) and p= f(a)) a Find the area of the region bounded by the astroid x = 9 cos ³t, 3 y = 9 sin ³t, for 0 st≤ 2.
Suppose the function y =h(x) is nonnegative and continuous on [x.³], which implies that the area bounded by the graph of h and the x-axis on [a.B] equals h(x) dx or ay dx. If the graph of y = h(x) on [a.B) is traced exactly once by the parametric equations x = f(t), y = g(t), for a ≤t≤ b, then it follows by substitution that the area bounded by his given by the equation below. ["h(x) dx = [y dx = [g(t) f' (t) dt, if œ=f(a) and ß=f(b) (or √n(x) dx =) ) dx =g(t) f'(t) dt, if x = f(b) and p= f(a)) a Find the area of the region bounded by the astroid x = 9 cos ³t, 3 y = 9 sin ³t, for 0 st≤ 2.
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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![Suppose the function \( y = h(x) \) is nonnegative and continuous on \([ \alpha, \beta ]\), which implies that the area bounded by the graph of \( h \) and the x-axis on \([ \alpha, \beta ]\) equals \(\int_{\alpha}^{\beta} h(x) \, dx\) or \(\int_{a}^{b} y \, dx\). If the graph of \( y = h(x) \) on \([ \alpha, \beta ]\) is traced exactly once by the parametric equations \( x = f(t), \, y = g(t) \), for \( a \leq t \leq b \), then it follows by substitution that the area bounded by \( h \) is given by the equation below.
\[
\int_{\alpha}^{\beta} h(x) \, dx = \int_{a}^{b} g(t) \, f'(t) \, dt, \text{ if } \alpha = f(a) \text{ and } \beta = f(b)
\]
(or
\[
\int_{\beta}^{\alpha} h(x) \, dx = \int_{b}^{a} g(t) \, f'(t) \, dt, \text{ if } \alpha = f(b) \text{ and } \beta = f(a)
\]
Find the area of the region bounded by the astroid \( x = 9 \cos^3 t, \, y = 9 \sin^3 t \), for \( 0 \leq t \leq 2\pi \).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F14ca10a7-9a5a-4024-afbf-f0f142f43830%2F4aa8fed6-a1e6-40da-b4b4-7861014c334a%2Fl04ifex_processed.png&w=3840&q=75)
Transcribed Image Text:Suppose the function \( y = h(x) \) is nonnegative and continuous on \([ \alpha, \beta ]\), which implies that the area bounded by the graph of \( h \) and the x-axis on \([ \alpha, \beta ]\) equals \(\int_{\alpha}^{\beta} h(x) \, dx\) or \(\int_{a}^{b} y \, dx\). If the graph of \( y = h(x) \) on \([ \alpha, \beta ]\) is traced exactly once by the parametric equations \( x = f(t), \, y = g(t) \), for \( a \leq t \leq b \), then it follows by substitution that the area bounded by \( h \) is given by the equation below.
\[
\int_{\alpha}^{\beta} h(x) \, dx = \int_{a}^{b} g(t) \, f'(t) \, dt, \text{ if } \alpha = f(a) \text{ and } \beta = f(b)
\]
(or
\[
\int_{\beta}^{\alpha} h(x) \, dx = \int_{b}^{a} g(t) \, f'(t) \, dt, \text{ if } \alpha = f(b) \text{ and } \beta = f(a)
\]
Find the area of the region bounded by the astroid \( x = 9 \cos^3 t, \, y = 9 \sin^3 t \), for \( 0 \leq t \leq 2\pi \).
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