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

Calculate the definite integral

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 $|x^2-5x|$.

Evaluate the area of the region above the $x$-axis bounded by 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 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 value of $c \in [2.5, 7.5]$ such that the area between the curve $y = x^2 - 5x$ and the $x$-axis from $x = 2.5$ to $x = c$ is equal to the area from $x = c$ to $x = 7.5$.

Find the $x$-coordinates where the curve $y = x^2 - 5x$ meets the $x$-axis. Hence, calculate the total area bounded by the curve and the $x$-axis between $x = 0$ and $x = 7.5$.

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

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

Use the linearity of the integral to show

and then compute its value.

Use Simpson’s rule with $n=4$ to approximate the value of $\int_{2.5}^{7.5} (x^2-5x) \, dx$.