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Calculus/applying integrals
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Question
ok so i have two questions about applying integrals.
1) a rectangle of length l and width w is revolved about line L. find
volume of resulting solid or revolution.
the figure is weird.
L
| /\
| / \l
|- - \ - -\ - -
| d \ \
| \ / w
| \/
yeah the dotted line is supposed to be a distance d from the corner of
the rectangle and the dotted line goes through the diagonal of the
rectangle.
2)the graph of y=f(x)passes through the origin. the arc length of the
curve from (0,0) to (x,f(x)) is given by
(x
s(x)=| sqrt(1+e^t)dt
)0
identify the function f.
yeah thats supposed to be an integral sign.
thanks
1) I wouled extend the outsides of the triangle all the way to the axis. The problem is then find the volume of four different triangles rotated about the axis. The first (1) is using sides l and w, the second (2) uses the side opposit l and w, the third (3) uses the side opposite w and l and the fourth (4) uses the two opposite sides. The value of the integral is then (1) - (2) - (3) + (4). This is beause we don't have the volume generated by (2) and (3), but when both are subtracted, the volume generated by (4) is subtracted twice so it must be added back in again.
2) The formula for arc length of a line is given by the integral from the start to the end of sqrt(1 + y^2(x)).
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