Area between two curves

In an extreme sports park, a skate ramp is to be installed. The side profile of the ramp is defined by the curve $f(x) = x^2$ and is to be covered with a safety tarp reaching the horizontal platform defined by the line $g(x) = 4$.

Calculate the area of the side section between the ramp and the horizontal platform (units in metres).

In an environmental engineering project, the cross section of an irrigation channel is studied. The profile of the channel is given by the curve $f(x) = x^2 - 1$ and the maximum water level is bounded by the line $g(x) = 3$.

Calculate the area of the channel's cross section that lies below the water level.

Two tech companies are comparing their user growth during their first year. The number of users (in thousands) of Company A follows the function $f(x) = x^2$, while Company B follows the function $g(x) = x$.

Calculate the area of the region between both graphs in the interval $[0, 1]$ to visualise the total accumulated difference in users during that first year.

Determine the area of the region bounded by the line $y = 2x$ and the curve $y = x^2 - 3$.

To do so, first find the points of intersection by setting both functions equal to each other and apply Barrow's rule to the integrand of the difference function.

Find the area of the region bounded by the graphs of the functions $f(x) = x^2 - 2x + 1$ and $g(x) = -x^2 + 4x + 1$. Determine which of the two functions lies above the other in the interval of integration by analysing the sign of the difference function $h(x) = f(x) - g(x)$.

Determine the area of the region enclosed by the cubic function $f(x) = x^3 - 4x$ and the line $g(x) = 0$ (the x-axis). Keep in mind that the region consists of two separate areas and that you will need to calculate the area of each one separately using absolute values if necessary.

Consider two functions representing the airflow in a wind tunnel during a 10-second experiment: $f(x) = x^3 - 6x^2 + 9x$ and $g(x) = x^2 - 3x$. The engineers need to know the total "interference volume", which is equivalent to the area enclosed between both graphs across their entire domain. Find the points of intersection between the two functions and calculate the total area of the regions they bound, taking into account that the functions may switch positions (which one is above the other) in the different intervals.

An architect is designing a decorative window whose silhouette is bounded by the modulus function $f(x) = |x^2 - 4|$ and the horizontal line $g(x) = 2$. Due to the complexity of the shape, the glass must be cut into several pieces. Calculate the total area of the window, correctly identifying the limits of integration and the functions acting as upper and lower boundaries in each section.

Given the function $f(x) = x^2 - 4$, find the equation of the tangent line to its graph at the point with x-coordinate $x = 3$. Once obtained, calculate the area of the region bounded by the graph of $f(x)$, the tangent line found, and the y-axis ($OY$).

Calculate the value of the positive real parameter $k$ such that the area of the region bounded by the parabola $y = x^2$ and the line $y = kx$ is equal to $36$ square units. Hint: Express the area in terms of $k$ and equate the result to the given value to solve for the unknown.

In a microeconomics study, the supply function for a product is $S(x) = x^2 + 2x + 1$ and the demand function is $D(x) = 11 - x^2$, where $x$ is the number of units in thousands. "Consumer Surplus" is defined as the area bounded by the demand curve, the horizontal line passing through the equilibrium point (where supply equals demand), and the y-axis. Calculate this surplus by first finding the equilibrium point in the first quadrant.