Chapter 12.1, Problem 105E

Area under a curve Suppose the function y = h(x) is nonnegative and continuous on [α, β], which implies that the area bounded by the graph of h and x-axis on [α, β] equals ${\displaystyle {\int }_{\alpha }^{\beta }h(x)}dx$ or ${\displaystyle {\int }_{\alpha }^{\beta }y}dx$. If the graph of y = h(x) on [α, β] 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 h is

${\displaystyle {\int }_{\alpha }^{\beta }h(x)}dx={\displaystyle {\int }_{\alpha }^{\beta }ydx={\displaystyle {\int }_a^{b}g(t)f^{\prime }\left(t\right)dt\text{ if }\alpha =f\left(a\right)\text{ and }\beta =f\left(b\right)}}$

$\left(\text{or }{\displaystyle {\int }_{\alpha }^{\beta }h(x)}dx={\displaystyle {\int }_a^{b}g(t)f^{\prime }\left(t\right)dt\text{ if }\alpha =f\left(b\right)\text{ and }\beta =f\left(a\right)}\right)$.

105. Find the area of the region bounded by the asteroid x = cos³ t, y = sin³ t, for 0 ≤ t ≤ 2π (see Example 8, Figure 12.17).