Question

A 1.5-mm layer of paint is applied to one side of the following surfaces. Find the approximate volume of paint needed. Assume x and y are measured in meters.

The spherical zone generated when the curve y=√8x−x^2 on the interval 1≤x≤7 is revolved about the x-axis

Accepted Answer

Step 1: Recognize that the problem involves finding the volume of paint applied to a surface generated by revolving a curve about the x-axis. This requires using the formula for the volume of a solid of revolution. Specifically, the volume of the spherical zone can be approximated using the shell method or disk method. Step 2: Write the formula for the volume of a solid of revolution using the disk method: V = π ∫[a,b] $$(y^2) dx$$, where y is the function describing the curve, and [a, b] is the interval of revolution. Here, y = √(8x − $$x^2) $$and the interval is [1, 7]. Step 3: Substitute the given function y = √(8x − $$x^2) $$into the formula. The integral becomes: V = π ∫[1,7] (8x − $$x^2) dx. $$Simplify the integrand by squaring the function. Step 4: Compute the integral ∫[1,7] (8x − $$x^2) dx. $$Break it into two parts: ∫[1,7] 8x dx and ∫[1,7] $$x^2 dx. $$Use the power rule for integration: ∫x^n dx = (x^(n+1))/(n+1). Step 5: After finding the volume of the solid of revolution, multiply the result by the thickness of the paint layer (1.5 mm = 0.0015 m) to approximate the volume of paint needed. This step involves scaling the surface area by the thickness of the paint layer.

A 1.5-mm layer of paint is applied to one side of the following surfaces. Find the approximate volume of paint needed. Assume x and y are measured in meters.

The spherical zone generated when the curve y=√8x−x^2 on the interval 1≤x≤7 is revolved about the x-axis

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A 1.5-mm layer of paint is applied to one side of the following surfaces. Find the approximate volume of paint needed. Assume x and y are measured in meters. The spherical zone generated when the curve y=√8x−x^2 on the interval 1≤x≤7 is revolved about the x-axis

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A 1.5-mm layer of paint is applied to one side of the following surfaces. Find the approximate volume of paint needed. Assume x and y are measured in meters.

The spherical zone generated when the curve y=√8x−x^2 on the interval 1≤x≤7 is revolved about the x-axis