Question Type 1: Area enclosed by the y-axis and a function Exercises

Determine the area under the curve $y=\frac{e^x}{2}$ between $x=-1$ and the $y$-axis.

Find the average value of $y=\frac{e^x}{2}$ on the interval $[0,2]$.

Find the value of $a$ such that the area under $y=\frac{e^x}{2}$ from $x=0$ to $x=a$ equals 1.

Find the area enclosed by the x-axis, the curve $y=\frac{e^x}{2}$, and the lines $x=1$ and $x=3$.

An engineer requires the cumulative area under $y=\frac{e^x}{2}$ from $x=0$ to $x=L$ to be at least 5. Find the minimal $L$.

Find the area of the region enclosed by the curve $y = \frac{e^x}{2}$, the $x$-axis, the $y$-axis and the line $x=1$.

The curve $y = \frac{e^x}{2}$ intersects the horizontal line $y=k$ at a point $P$. The region $R$ is bounded by the curve, the coordinate axes, and the vertical line passing through $P$.

Given that the area of $R$ is $1$, find the value of $k$.

Determine $b>0$ such that the average value of $y=\frac{e^x}{2}$ on $[0,b]$ is 1.

Find the area of the region bounded by the y-axis and the curve $y=\frac{e^x}{2}$ between $y=1$ and $y=2$.

Calculate the area of the region enclosed by the curve $y = \frac{e^x}{2}$, the $x$-axis, and the vertical lines $x=0$ and $x=2$.