Now we can say that the volume of the solid created by rotating the region under y = 4e−x2and above the x-axis between x = 0 and x = 1 around the y-axis isV = b2𝜋rh dxa= 2𝜋x
Question
Now we can say that the volume of the solid created by rotating the region under y = 4e−x2and above the x-axis between x = 0 and x = 1 around the y-axis isV = b2𝜋rh dxa= 2𝜋x
Solution
It seems like you're trying to find the volume of a solid of revolution using the method of cylindrical shells. The formula for the volume of a cylindrical shell is V = ∫[a, b] 2πrh dx, where r is the radius of the shell (the x-coordinate), h is the height of the shell (the function value), and dx is the thickness of the shell.
Given the function y = 4e^(-x^2), the limits of integration from x = 0 to x = 1, and the rotation around the y-axis, we can substitute these values into the formula:
V = ∫[0, 1] 2πx * 4e^(-x^2) dx
This integral can be solved using methods of integration, such as substitution or integration by parts.
Similar Questions
Step 2Now we can say that the volume of the solid created by rotating the region under y = 4e−x2and above the x-axis between x = 0 and x = 1 around the y-axis isV = b2𝜋rh dxa= 1 1 2𝜋x $$4e−x2 dx.0Step 3The integral 2𝜋 4xe−x2 dx can be done with the substitution u = $$−x2 and du = $$−2x dx.Step 4With the substitution, we have 2𝜋 4xe−x2 dx = − $$4π eu du = − $$4πeu + C.Step 5Going back to x, the volume of our solid is
Use the method of cylindrical shells to find the volume V generated by rotating the region bounded by the given curves about the y-axis.y = 4e−x2, y = 0, x = 0, x = 1Sketch the region and a typical shell.Step 1Rotating a vertical strip around the y-axis creates a cylinder with radius r = $$x and height h = $$4e−x2 . Sketch the region and a typical shell. Step 2Now we can say that the volume of the solid created by rotating the region under y = 4e−x2and above the x-axis between x = 0 and x = 1 around the y-axis isV = b2𝜋rh dxa= 2𝜋x dx.
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