Use the general slicing method to find the volume of the follo...

Question

Use the general slicing method to find the volume of the following solids.

The solid whose base is the region bounded by the semicircle y=√1−x^2 and the x-axis, and whose cross sections through the solid perpendicular to the x-axis are squares

semi

Accepted Answer

Determine the volume of the solid whose base is the region R bounded by the semicircle, Y equals the square root of 4 minus X2, and the X-axis, and whose cross sections perpendicular to the X-axis are equilateral triangles. Use the general sicing method. We're also given a graph of our region as shown below. Now, to solve this, we will first take our integral. The integral for the volume is given by the integral from A to B. Of the area in terms of XDX. Now, we need to know what the area of an equilateral triangle is. So you have area of an equal level little triangle. Which is given by the equation square root of 3 divided by 4, side length squared. In this case, now we will find side length. This is given by the difference of our topmost equation or our lower equation. S will be equals to the square root of 4 minus X squared after the topmost equation. Minus the x-axis, which is Y equals 0. This will just be the square root of 4 minus X2d. Now we can go ahead and write our integral. We have V equals the integral from -2 to 2 of the square root of 3 divided by 4. Square root of 4 minus X2 squared. DX We now have V equals the integral from -2 to 2 of the square root of 3/4 multiplied by 4 minus X 2 DX by removing the radical. Now we can solve our integral. This will be given by The square root of 3/4. Multiplying By the integral from -2 to 2 of 4. DX Plus In this case is actually minus because of the negative on the X2. Minus the interval from -2 to 2 of X 2 DX. This will give us the square root of 3 divided by 4. Multiplied by 4 X from 2 to 2. Minus XQ divided by 3 from 2 to 2. And from there, we can plug in our values. We plug in our values, we will get the square root of 3 divided by 4. Of In this case, our first value gives us 16. And our second value GSS 16 divided by 3. This simplifies the square root of 3 divided by 4, multiplied. By 48 divided by 3, minus 16 divided by 3. Which gives us the square root of 3, divided by 4, multiplied by 32/3. We can finally simplify. By multiplying together, to give us a value of 8 square of 3 divided by 3. This means our answer is 8 square of 3 divided by 3. Cubic units. OK, I hope to help you solve the problem. Thank you for watching. Goodbye.

Use the general slicing method to find the volume of the following solids.The solid whose base is the region bounded by the semicircle y=√1−x^2 and the x-axis, and whose cross sections through the solid perpendicular to the x-axis are squares

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Use the general slicing method to find the volume of the following solids.
The solid whose base is the region bounded by the semicircle y=√1−x^2 and the x-axis, and whose cross sections through the solid perpendicular to the x-axis are squares