Textbook Question
Let R be the region bounded by the curve y=√cos x and the x-axis on [0, π/2]. A solid of revolution is obtained by revolving R about the x-axis (see figures).
c. Write an integral for the volume of the solid.
9. Graphical Applications of Integrals
Introduction to Volume & Disk Method
2:43 minutesProblem 6.4.72a
Textbook Question
Explain the steps required to find the length of a curve x = g(y) between y=c and y=d.
Verified step by step guidance1
Identify the function given as $x = g(y)$ and the interval over which you want to find the curve length, from $y = c$ to $y = d$.
Recall the formula for the length of a curve expressed as $x = g(y)$, which is given by the integral:
$$L = \int_{c}^{d} \sqrt{1 + \left(\frac{dx}{dy}\right)^2} \, dy$$
Compute the derivative $\frac{dx}{dy}$ by differentiating $g(y)$ with respect to $y$.
Substitute $\frac{dx}{dy}$ into the integral formula to get:
$$L = \int_{c}^{d} \sqrt{1 + \left(g'(y)\right)^2} \, dy$$
Evaluate the integral over the interval $[c, d]$ to find the length of the curve.
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Key Concepts
Here are the essential concepts you must grasp in order to answer the question correctly.
Arc Length Formula for Parametric Curves
The arc length of a curve defined by x = g(y) between y = c and y = d is found by integrating the square root of 1 plus the derivative of x with respect to y squared. This formula accounts for the infinitesimal distances along the curve, summing them to find the total length.
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Derivative of the Function x = g(y)
To apply the arc length formula, you need the derivative dx/dy, which measures how x changes with respect to y. This derivative is essential because it determines the slope of the curve and influences the length calculation by affecting the integrand.
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Definite Integration over the Interval [c, d]
After setting up the integrand involving the derivative, you compute the definite integral from y = c to y = d. This integration sums the infinitesimal arc lengths along the curve, yielding the total length between the specified bounds.
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