Find the lengths of the curves in Exercises 1-12 . If you have graphing software, you may want t

step 1 So for this problem, we want to find the length of a curve. So we want to use our arc length for

Find the lengths of the curves in Exercises 1-12 . If you...

Key Concepts

Integration Techniques

Integration techniques are the methods used to solve integrals that arise in applications like arc length calculations. Depending on the integrand, techniques such as substitution, simplification, and recognizing standard integral forms may be necessary. Mastery of these techniques allows one to handle the complex expressions that result from involving the derivative squared in the square root of the arc length formula.

Arc Length Formula

The arc length formula is a fundamental concept in calculus used to calculate the distance along a curve. For a function y = f(x) defined on an interval [a, b], the arc length L is given by the integral L = ? from a to b ?(1 + (dy/dx)²) dx. This formula essentially sums the lengths of infinitesimally small straight line segments that approximate the curve, making use of the Pythagorean theorem to combine horizontal and vertical changes.

Differentiation of Functions

Differentiation is the process of finding the derivative of a function, which represents the rate at which the function’s value changes with respect to changes in its input. In the context of computing arc lengths, finding the derivative dy/dx is critical because it appears in the integrand of the arc length formula. The derivative of functions, particularly power functions, is calculated using rules such as the power rule, which simplifies the integration process.

Definite Integration

Definite integration is used to compute the accumulated value of a function over a specific interval. In arc length problems, after setting up the integral with the arc length formula, the definite integral is evaluated between the given limits. This involves calculating the antiderivative and applying the Fundamental Theorem of Calculus to find the exact length of the curve.

Transcript

00:01 So for this problem, we want to find the length of a curve.

00:04 So we want to use our arc length formula, which is given by the integral from a to b, of the square root of 1 plus our derivative squared.

00:26 So for this problem, we're given the function y is equal to x to the three halves, a is equal to zero, and b is equal to so computing our derivative, we see d -y -d -x, well that's just going to be three - halves x to the one -half power.

00:52 So what we want to solve is the integral from 0 to 4, the square root of 1 plus 9 over 4x d x and the 9 over 4x comes from squaring our derivative.

01:10 So we can't do this right away.

01:13 We have to do u -substubes.

01:14 So we notice that the derivative of the inside of our square root is a constant.

01:23 It has no x's.

01:24 So we want to use that as our u.

01:32 We take the derivative, d u d x, well that's going to be equal to 9 over 4.

01:40 And we want to get dx by itself so that we can substitute, and we get that 4 ninth d u is equal to d x.

01:51 I also want to change the bounds of the integral.

01:54 So we evaluate u for our two x values.

01:58 So u when x is equal to zero, this is going to be 1 plus 9 over 4 times 0, that's equal to 1.

02:10 And then u of 4 is 1 plus 9 over 4 times 4, and that's equal to 10...

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