Applications of Integration

Given that two functions intersect at points $(a,b)$ and $(c,d)$ with $a < c$, which integral represents finding area between these curves from leftmost to rightmost intersection point?

Find the area between the curves $y = e^{2x}$ and $y = 2e^x$ for $0 \\leq x \\leq \\ln(2)$.

Find the area between the curves $y = x^2$ and $y = 2x$ for $0 \leq x \leq 2$.

Find the area between the curves $y = e^x$ and $y = x$ for $0 \leq x \leq 1$.

What value of k would make continuous on [-1, 1] if $f(x) = \begin{cases} k + x & \text{for } x < 0 \\ \sin\left(\frac{\pi}{k} x\right) & \text{for } x \geq 0 \end{cases}$?

Determine the area enclosed by the curves $y = \sin(x)$ and $y = \cos(x)$ for $0 \leq x \leq \frac{\pi}{2}$.

Determine the area enclosed by the curves $y = x^3$ and $y = 3x$ for $-1 \\leq x \\leq 1$.

If you set up an integral to find the area between $\sqrt{x}$ and $\frac{x}{2}$ from $x=1$ to $x=4$, which function should be on top in your integral setup?

What is the area of the region enclosed by $y = x^2$ and $y = 2x$, between $x=0$ and their intersection point?

Determine the area enclosed by the curves $y = x^2$ and $y = \sqrt{x}$ for $0 \leq x \leq 1$.