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Each integral represents the volume of a solid. Describe the solid.

$ \pi \displaystyle \int_{0}^\pi \sin x dx $

$\pi \int_{0}^{\pi} \sin x d x=\pi \int_{0}^{\pi}(\sqrt{\sin x})^{2} d x$ describes the volume of solid obtained by rotating the region

$g=\{(x, y) | 0 \leq x \leq \pi, 0 \leq y \leq \sqrt{\sin x}\}$ of the $x y$ -plane about the $x$ -axis.

Applications of Integration

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Missouri State University

Harvey Mudd College

University of Michigan - Ann Arbor

in this exercise, we're being asked to look at this expression for a volume of a solid in integral form and describe the solid that is being the solid whose volume we are calculating. And the first thing that I noticed is the dX is giving me the width of individual slices. So I need to use the other two pieces to determine what shape the area of those slices are. Well, sine of X. Yeah is the only thing that's changing and sine of X is going to give me Going from 0 to Pi. Just that sort of it will shape and it's being multiplied by a constant. So I've got something that is changing being multiplied by a constant, an individual value times a constant as an area. Two numbers being multiplied together to give me an area that's just a rectangle. So this is telling me that I've got rectangular slices. Mhm With an infinite decimal with delta X. And the one dimension of those slices is going is dependent on the X value and at zero it's going to be zero anywhere along the rest of the base. It's going to have a height that is given by that curve sign X. But the width is going to be a constant pi So if I have a slice there the way that will be pie. If I have a slice here, the width will be pie, I have a slice here 20 ft high. And if I have a slice at that end, the width will be pie. So what I have is a shape that has a rectangular base that has a width of pie and the length of pie. So it's actually a square base and it's being cut into rectangular slices. The height of the slices is given by the function sign over pie. The width of the slices is the width of the square pie. And what I've got overall is the square base here. The height is zero here, flight zero In the middle. The height is one and the shape of the face is making or the shape the height of the shape is making this sort of silo type sheep. Yeah, I can describe that verbally has oh a square base. This links hi and bring home parallel angular. It's like this make up the area. All right. All right. You spell directly of the 10 you were uh slices is given by X. X. Yeah. And that describes this shape mathematically visually. The shape has a square base and is making a silo shape. The ends of the silo are that sine of pi curve. So this total light is one is total with this pie. This total with this pie. This total with his pie. Uh huh.

University of Southern California

Applications of Integration