Find the area of the surface generated when the given curve is revolved about the given axis.


y=4x−1, for 1≤x≤4; about the y-axis (Hint: Integrate with respect to y.)

Verified step by step guidance

1. Step 1: Recall the formula for the surface area of a curve revolved about the y-axis. The formula is: $A\; =\; 2\pi \; \int \; x\; \surd (1\; +\; (dx/dy)²)\; dy$, where x is expressed as a function of y.
2. Step 2: Rewrite the given equation y = 4x - 1 to express x in terms of y. Solve for x: $x\; =\; (y\; +\; 1)/4$.
3. Step 3: Compute $dx/dy$ by differentiating x = (y + 1)/4 with respect to y. The derivative is: $dx/dy\; =\; 1/4$.
4. Step 4: Determine the limits of integration for y. Since the curve is defined for $1\; \le \; x\; \le \; 4$, substitute these x-values into the equation y = 4x - 1 to find the corresponding y-values. For x = 1, y = 3; for x = 4, y = 15. Thus, the limits of integration are $3\; \le \; y\; \le \; 15$.
5. Step 5: Substitute x = (y + 1)/4 and dx/dy = 1/4 into the surface area formula. The integral becomes: $A\; =\; 2\pi \; \int$₃¹⁵ ((y + 1)/4) √(1 + (1/4)²) dy. Simplify the expression inside the integral and proceed to evaluate the integral.

Key Concepts

Surface Area of Revolution

The surface area of revolution refers to the area of a three-dimensional surface created when a two-dimensional curve is rotated around an axis. To find this area, we typically use integral calculus, applying the formula that involves the arc length of the curve and the radius of rotation. In this case, since the curve is revolved around the y-axis, we will need to express the curve in terms of y.

Changing Variables in Integration

Changing variables in integration is a technique used to simplify the process of evaluating integrals. In this problem, we need to integrate with respect to y, which requires us to express x as a function of y. This often involves solving the original equation for y and determining the limits of integration based on the given x-interval.

Definite Integrals

Definite integrals are used to calculate the area under a curve between two specified limits. In this context, we will evaluate the integral of the surface area formula over the interval defined by the y-values corresponding to x = 1 and x = 4. Understanding how to set up and compute definite integrals is crucial for finding the total surface area generated by the revolution of the curve.