The graph of g consists of two straight lines and a semicircle. Use it to eveluate the integral. 12210203861.jpg int_0^35 g(x)dx

Question
Applications of integrals
asked 2021-02-19
The graph of g consists of two straight lines and a semicircle. Use it to eveluate the integral.
image
\(\displaystyle{\int_{{0}}^{{35}}}{g{{\left({x}\right)}}}{\left.{d}{x}\right.}\)

Answers (1)

2021-02-20
The integral \(\displaystyle{\int_{{0}}^{{35}}}{g{{\left({x}\right)}}}{\left.{d}{x}\right.}\) is the area between the graph of g and the horizontal x-axis on the interval \(\displaystyle{0}\le{x}\le{35}\). We separate this integral into the parts of (a) and (b) along with
\(\displaystyle{\int_{{30}}^{{35}}}{g{{\left({x}\right)}}}{\left.{d}{x}\right.}\).
\(\displaystyle{\int_{{30}}^{{35}}}{g{{\left({x}\right)}}}{\left.{d}{x}\right.}\) represents the area that forms a triangle with base b = 5 and height h = 5.
The area of a triangle is the product of the base b and the height h, divided by 2.
\(\displaystyle{\int_{{30}}^{{35}}}{g{{\left({x}\right)}}}{\left.{d}{x}\right.}=\frac{{{b}\cdot{h}}}{{2}}\)
\(\displaystyle=\frac{{{5}\cdot{5}}}{{2}}\)
\(\displaystyle=\frac{{25}}{{2}}\)
=12.5
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