⧼exchistory⧽
13 exercise(s) shown, 0 hidden
BBy Bot
Nov 03'24
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[/math]
In each of the following, draw the curve defined by the equation [math]r = f(\theta)[/math] in polar coordinates. Show the region [math]R[/math] bounded by the curve and the lines [math]\theta = a[/math] and [math]\theta = b[/math], and compute its area.
- [math]r=4\cos \theta[/math], [math]a=0[/math] and [math]b=\frac\pi2[/math].
- [math]r=3(1+\cos\theta)[/math], [math]a=0[/math] and [math]b=\pi[/math].
- [math]r=3(1+\sin\theta)[/math], [math]a=0[/math] and [math]b=\frac\pi2[/math].
- [math]r=\frac2{\cos\theta}[/math], [math]a=-\frac\pi4[/math] and [math]b=\frac\pi4[/math].
BBy Bot
Nov 03'24
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[/math]
For each of the following equations [math]r=f(\theta)[/math] and pairs of numbers [math]a[/math] and [math]b[/math], draw the region [math]R[/math] consisting of all points with polar coordinates [math](r,\theta)[/math] such that [math]a\leq\theta\leq b[/math] and [math]0\leq r\leq f(\theta)[/math]. Compute [math]\mbox{''area''}(R)[/math].
- [math]r=4\sin\theta[/math], [math]a=0[/math] and [math]b=\pi[/math].
- [math]r=\frac4{\sin\theta}[/math], [math]a=\frac\pi4[/math] and [math]b=\frac{3\pi}4[/math].
- [math]r=2\theta[/math], [math]a=\pi[/math] and [math]b=2\pi[/math].
- [math]r=\frac1{2\cos\theta+3\sin\theta}[/math], [math]a=0[/math] and [math]b=\frac\pi2[/math].
- [math]r=\sqrt{2\cos2\theta}[/math], [math]a=0[/math] and [math]b=\frac\pi4[/math]. (See Example)
BBy Bot
Nov 03'24
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[/math]
Identify and draw the curve defined by the equation [math]r=4\sin\theta[/math] in polar coordinates, and show the region [math]R[/math] bounded by the curve. Is it true in this case that
[[math]]
\mbox{''area''}(R) = \frac12
\int_0^{2\pi} r^2 \; d\theta
?
[[/math]]
Explain your answer.
BBy Bot
Nov 03'24
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[/math]
Each of the following curves, defined by an equation [math]r=f(\theta)[/math] in polar coordinates, bounds a region [math]R[/math] in the plane. Draw the curve and find the area of [math]R[/math].
- [math]r=a(1+\cos\theta)[/math], [math]a \gt 0[/math]
- [math]r=a(1+\sin\theta)[/math], [math]a \gt 0[/math]
- [math]r=5[/math]
- [math]r=2+\cos\theta[/math]
- [math]r=4\sin\theta[/math]
- [math]r=-4\cos\theta[/math].
BBy Bot
Nov 03'24
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[/math]
The curve defined by the equation [math]r=\frac1{1+\cos\theta}[/math] in polar coordinates is a parabola similar to the one discussed in Example.
- Draw the parabola, and show the region [math]R[/math] bounded by this curve and the line [math]\theta=\frac\pi2[/math].
- Express [math]\mbox{''area''}(R)[/math] as a definite integral using the integral formula for area in polar coordinates.
- Evaluate the integral in part (b) using the trigonometric substitution [math]z=\tan \frac\theta2[/math] (see equation) and the Change of Variable Theorem for Definite Integrals.
- Write this curve as the graph of an equation in [math]x[/math]- and [math]y[/math]-coordinates, and thence compute area(R).
BBy Bot
Nov 03'24
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[/math]
Find the area of the region which lies between the two loops of the limaçon [math]r=1+2\cos \theta[/math].
BBy Bot
Nov 03'24
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[/math]
Find the area of the region bounded by the lemniscate [math]r^2 = 2a^2 \cos 2\theta[/math].
BBy Bot
Nov 03'24
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[/math]
Find the area [math]A[/math] of the region which lies inside the cardioid [math]r=2(1+\cos\theta)[/math] and outside the circle [math]r=3[/math].
BBy Bot
Nov 03'24
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[/math]
The region [math]R[/math] bounded by the cardioid [math]r=4(1+\sin\theta)[/math] is cut into two regions [math]R_1[/math] and [math]R_2[/math] by the polar graph of the equation [math]r=\frac3{\sin\theta}[/math]. Compute the areas of [math]R[/math], [math]R_1[/math], and [math]R_2[/math].
BBy Bot
Nov 03'24
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[/math]
Find the arc length of the cardioid defined by the equation [math]r=a(1+\cos\theta)[/math], where [math]a[/math] is an arbitrary positive constant.