︠d9f69ebc-3f22-4ce9-a60d-df3bf8512dd3i︠
%hide
var('x')
pf = plot(x^2, x, 0, 1, thickness = 2, color = 'red')
pg = plot(2*x - x^2, x, 0, 1, fill = x^2, fillcolor = 'yellow', color = 'blue', thickness = 2)
html("Section 7.1 Example 2: Area between $y = x^2$ and $y = 2x - x^2$")
show(pf + pg)
html("Area = $\int_0^1 \\left[f(x) - g(x)\\right]\\,dx = %s$" %latex(integrate(2*x - 2*x^2, x, 0, 1)))
︡1ae8aa7b-dd7c-4523-83e1-5ebb1c86b87d︡{"html": "Section 7.1 Example 2: Area between y = x^2 and y = 2x - x^2"}︡{"html": "<img src='cell://sage0.png'>"}︡{"html": "Area = \\int_0^1 \\left[f(x) - g(x)\\right]\\,dx = \\frac{1}{3}"}︡
︠c7316a63-6d9f-4bf9-964e-6464be7e597ci︠
%hide
var('t')
html("Section 7.2 Example 3: Rotating region bounded by $y = \\sqrt{x}$, $y = 8$, and $y = 0$ about the $y$-axis")
p = parametric_plot3d([x, sqrt(x)*cos(t), sqrt(x)*sin(t)], (x, 0, 1), (t, 0, 2*pi))
show(p, mesh = True, aspect_ratio = 1)
html("Volume = $%s$" %latex(integrate(pi*x, x, 0, 1)))
︡3c994fe0-df6d-4b2a-ad5c-694c397109fc︡{"html": "Section 7.2 Example 3: Rotating region bounded by y = \\sqrt{x}, y = 8, and y = 0 about the y-axis"}︡{"html": "Volume = \\frac{1}{2} \\, \\pi"}︡

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