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

\(\displaystyle{\int_{{30}}^{{35}}}{g{{\left({x}\right)}}}{\left.{d}{x}\right.}\).

\(\displaystyle{\int_{{30}}^{{35}}}{g{{\left({x}\right)}}}{\left.{d}{x}\right.}\) 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