41–48. Geometry problems Use a table of integrals to solve the following problems.
42. Find the length of the curve y = x^(3/2) + 8 on the interval from 0 to 2.

Verified step by step guidance

1. Step 1: Recall the formula for the length of a curve. The length of a curve y = f(x) from x = a to x = b is given by: ${\int }_a^b\sqrt{{\frac{\mathrm{dy}}{\mathrm{dx}}}^2+1}\mathrm{dx}$. This formula accounts for both the vertical and horizontal changes along the curve.
2. Step 2: Compute the derivative of y = x^(3/2) + 8 with respect to x. The derivative is: $\frac{\mathrm{dy}}{\mathrm{dx}}=\frac{3}{2}{x}^{\frac{1}{2}}$. The constant 8 disappears because its derivative is 0.
3. Step 3: Substitute the derivative into the curve length formula. The integrand becomes: $\sqrt{{\frac{3}{2}{x}^{\frac{1}{2}}}^2+1}$. Simplify the square of the derivative term.
4. Step 4: Simplify the integrand further. The square of the derivative term becomes: $\frac{9}{4}x$. Thus, the integrand is: $\sqrt{\frac{9}{4}x+1}$.
5. Step 5: Use a table of integrals to evaluate the integral of $\sqrt{\frac{9}{4}x+1}$ from x = 0 to x = 2. Look for a matching formula in the table of integrals and apply it to compute the curve length.

Key Concepts

Arc Length Formula

The arc length formula is used to calculate the length of a curve defined by a function. For a function y = f(x), the length L from x = a to x = b is given by the integral L = ∫[a to b] √(1 + (dy/dx)²) dx. This formula incorporates the derivative of the function to account for the slope of the curve, providing an accurate measure of its length.

Differentiation

Differentiation is the process of finding the derivative of a function, which represents the rate of change of the function with respect to its variable. In the context of the arc length formula, the derivative dy/dx is crucial as it helps determine how steep the curve is at any point, affecting the overall length calculation.

Definite Integrals

Definite integrals are used to calculate the accumulation of quantities, such as area under a curve, over a specific interval [a, b]. In the context of finding arc length, the definite integral computes the total length of the curve between two points by summing infinitesimally small segments of the curve, providing a precise measurement of its extent.