Question Type 3: Calculating the area between a function and the y-axis Exercises

Calculate the definite integral

Use the linearity of the integral to show

and then compute its value.

Find the area enclosed by the curve $y=x^2-5x$, the $x$-axis, and the vertical lines $x=2.5$ and $x=7.5$.

Find the average value of $f(x)=x^2-5x$ on $[2.5,7.5]$ and verify that area $=f_{\text{avg}}\times(7.5-2.5)$.

Evaluate the area above the $x$-axis between the curve $y=x^2-5x$ and the lines $x=5$ and $x=7.5$.

Compute the area of the region where the curve $y=x^2-5x$ lies below the $x$-axis, between $x=2.5$ and $x=5$.

Find the $x$-coordinates where $y=x^2-5x$ meets the $x$-axis, then compute the total area trapped between the curve and the $x$-axis from $x=0$ to $x=7.5$.

Determine the point $c\in[2.5,7.5]$ guaranteed by the Mean Value Theorem for Integrals so that $\int_{2.5}^{7.5}(x^2-5x)\,dx = (7.5-2.5)\bigl(c^2-5c\bigr).$

Use Simpson’s rule with $n=4$ to approximate the area under $y=x^2-5x$ from $x=2.5$ to $x=7.5$.

Evaluate the total (unsigned) area between the curve $y=x^2-5x$ and the $x$-axis from $x=2.5$ to $x=7.5$ by integrating $|x^2-5x|$.

Find $c\in[2.5,7.5]$ such that the area under $y=x^2-5x$ from $2.5$ to $c$ equals the area from $c$ to $7.5$.