14.02 Area between curves
For each of the following pairs of functions:
Graph the functions on the same set of axes.
Hence, find the area of the region bounded by the functions and the x-axis.
x + y = 2 and y = x.
x + y = 6 and y = 2x.
For each of the following pairs of functions:
Graph the functions on the same set of axes.
State the value(s) of x at which the two functions intersect.
Hence, find the area of the region bounded by the functions and the x-axis.
y=2x and y=(x-4)^2.
y = x^{2} and y = \left(x - 2\right)^{2}
Find the area of the region bounded by the following pairs of functions and the x-axis:
y=x^2 - 4 and y=-2x+11.
y=x^3 and y=-3x+14.
y=(x+2)^2 and y=-4x-3.
Find the area of the region bounded between the curve y = \sqrt{x + 5}, the line y = - x - 3 and the x-axis:
Find the area of the region bounded between the curve y = \sqrt{3-x}, the line y=\dfrac{2x+8}{3} and the x-axis:
Find the area of the region bounded by the curves y = e^{ 2 x} and y = e^{ - x }, the x-axis, and the lines x = - 2 and x = 2:
Find the area of the region bounded by y=x^2 + 1, y=-\dfrac{x}{2}+6, the x-axis, and the y-axis.
The diagram shows the shaded region bounded by y = 3, y = 0, y = 6 x - x^{2} - 8, x = 0 and x = 6:
Find the area of the shaded region.
Consider the following graph:
Find the area of the shaded region B.
Find the area of the shaded region A.
What is the ratio of the areas A:B?
Consider the following graph:
Find k such that the two shaded regions have equal area.
Find the exact area between the graph of y = \sqrt{4 - x^{2}} and the lines x = 2 and y = 2.
Consider the functions y = e^{\frac{x}{6}} and y = e^{\frac{3}{2}}.
Find the x-coordinate of the point of intersection.
Calculate the exact area bounded by y = e^{\frac{x}{6}}, the y-axis and the line y = e^{\frac{3}{2}}.
Consider the curve y = e^{x}.
Find the area bound by the curve, the x-axis, the y-axis, and the line x = 3.
Find the equation of the tangent to the curve y = e^{x} at the point where x = 3.
Find the exact area enclosed between y = e^{x}, the x-axis, the y-axis, and the tangent at x = 3.
Find the area enclosed between the lines \\y = 2 x, y = \dfrac{1}{3} x and x = 6:
The diagram shows the shaded region bounded by y = 4 - x^{2}, y = 1 - x^{2} and the x-axis:
Find the area of the shaded region.
Consider the graph of the functions y = x^{2} and y = x^{4}:
Find the area enclosed between the two curves.
Consider the graph of the functions \\ f\left(x\right) = x \left(x - 6\right)^{2} and g\left(x\right) = x^{2}:
State the values of x at which the curves intersect.
Hence, find the total area bounded between the curves.
Find the area bounded by the curves and the x-axis.
For each of the following pairs of functions:
Graph the functions on the same set of axes.
State the values of x at which the line and the curve intersect.
Hence, find the area enclosed between the line and the curve.
y = x^{2} and y = x + 2
y = x \left(x - 4\right) and y = x
y = - x^{2} + 8 and y = - x + 2
y = x \left(x - 4\right)^{2} and y = x
y = - x \left(x - 3\right)^{2} and y = - x
For each of the following pairs of functions:
Find the values of x at which the two curves intersect.
Find the area enclosed between the two curves.
y = x^{2} - 48 and y = - \left(x - 2\right)^{2} + 4
y = x and y = \left(x - 5\right)^{3} + 5
y = 4 x - 12 and y = x \left(x - 3\right)^{2}
y = 2 x and y = x^{2} - 15
Consider the functions y = - 2 x \left(x - 4\right) and y = - x + 4.
Graph the functions on the same set of axes.
State the values of x at which the curve and the line intersect.
Hence, find the area enclosed between the curves.
Find the small area enclosed between the curve, the line and the y-axis.
Consider the functions y = x^{2} and y = 8 - x^{2}.
Graph the functions on the same set of axes.
State the values of x at which the curves intersect.
Hence, find the area bounded between the curves.
Find the small area bounded between the curves and the x-axis, correct to one decimal place.
The diagram shows the graphs of y = \sin x and y = 3 \sin x:
Determine whether the following integral could be performed to calculate the area of the region bound by the curves, A.
\int_{0}^{\pi} \left( 3 \sin x - \sin x\right) dx
\int_{0}^{\pi} 3 \sin x dx - \int_{0}^{\pi} \sin x dx
\int_{0}^{\pi} 3 \sin x dx + \int_{0}^{\pi} \sin x dx
2 \int_{0}^{\pi} \sin x dx
\int_{0}^{\pi} \left( 3 \sin x + \sin x\right) dx
4 \int_{0}^{\pi} \sin x dx
Find the area of the shaded region A.
What is the ratio of the areas A:B?
Find the area of the region bounded by the curves y = \sin x and y = \cos x, and the lines x = 0 and x = \dfrac{\pi}{4}:
Find the area enclosed by the two curves of \\y = \sin 2 x and y = \cos x on the interval \\ \dfrac{\pi}{6} \leq x \leq \dfrac{\pi}{2}.
Find the area between the two curves of y = \sin 8 x and y = \cos 8 x on the interval \left[0,\dfrac{\pi}{16}\right].
Find the area enclosed by the two curves of f\left(x\right) = 5 \cos 2 x and g\left(x\right) = 5 \sin \left(\dfrac{x}{2}\right) between x = \dfrac{\pi}{5} and x = \pi. Round your answer to two decimal places.
Find the area of the region bounded by f\left(x\right) = 4 \cos x, g\left(x\right) = - 4 \cos \left(x - \dfrac{\pi}{2}\right), the y-axis and x = \dfrac{3 \pi}{4}.
Find the area of the region bounded by the line y = 2 \pi - x and the curve y = - 2 \sin x and the x-axis between x = 0 and x = \dfrac{2 \pi}{3}.
Consider the functions y = \sin x and y = \cos x.
Graph the functions on the same set of axes in the interval \left[0, \pi\right].
Hence, find the area between the two curves in this interval.
Consider the functions y = \sin 4 x and y = \cos 4 x for 0 \leq x \leq \dfrac{\pi}{2}.
Find the x-coordinates of the points of intersection of the two curves in the given interval.
Graph the two functions on the same set of axes.
Hence, find the area bounded by the two curves between the points of intersection found in part (a).
Consider the functions f\left(x\right)=\sin 4 x and g\left(x\right)=1 - \cos 4 x.
Graph the functions from 0 to \dfrac{\pi}{2} on the same set of axes.
For what values of x is f\left(x\right) = g\left(x\right)?
Evaluate \int_{0}^{\frac{\pi}{2}} \left(1 - \cos 4 x - \sin 4 x\right) dx.
Find the area between the two curves on the interval \left[0, \dfrac{\pi}{2}\right].
Find the exact area between the following graphs and lines:
y = x^{3} - 8, x = 2, and y = - 8
y = e^{x}, y = e^{ - x }, and x = 3
y = e^{ 5 x}, y = e^{ - 5 x }, x = - 2, and x = 2
y = x, y = e^{1 - x}, and x = 3
Consider the functions f \left( x \right) = e^{x} + 12 and g \left( x \right) = 4 x + e^{3}.
Find the x-coordinate of the point of intersection that is to the right of origin.
Find the area in the first quadrant bounded by the two curves and the y-axis.
Consider the graphs of f \left( x \right) = x^{2} and g \left( x \right) = \sqrt{x}:
Find the x-coordinates of the points of intersection.
Find the shaded area bounded by the two curves.
Complete the following table for the area bounded by f \left( x \right) = x^{n} and g \left( x \right) = \sqrt[n]{x}.
| n | 2 | 3 | 5 | 8 |
|---|---|---|---|---|
| \text{Area} |
Consider the graph below which has the following properties:
Functions f \left( x \right) and g \left( x \right) intersect at x = 0, x = 2 and x = 3
Region A has an area of 22 square units
Regions A and B have a combined area of 56 square units
\\ \int_{2}^{3} f \left( x \right) dx = - 18 and \int_{2}^{3} g \left( x \right) dx = - 13
Find \int_{0}^{2} f \left( x \right) dx.
Find \int_{0}^{2} \left(f \left( x \right) - g \left( x \right) \right) dx.
Find the area of region C.
Find the area of region D.
Find \int_{0}^{3} \left(f \left( x \right) - g \left( x \right) \right) dx.
Consider the graph below which has the following properties:
Functions f \left( x \right) and g \left( x \right) intersect on the x-axis at points p,q, and r
Region A has an area of 16\text{ units}^2
Regions A and B have a combined area of 26\text{ units}^2
- \int_{p}^{r} f \left( x \right) dx = 11 and \int_{p}^{r} g \left( x \right) dx = - 9
Find the value of \int_{p}^{q} f \left( x \right) dx.
Find the value of \int_{p}^{q} g \left( x \right) dx.
Find the area of region C.
Find the value of \int_{p}^{r} \left(f \left( x \right) - g \left( x \right) \right) dx.
Consider the graph of the functions f \left( x \right) = 3 x and g \left( x \right) = \dfrac{x^{2}}{2} together with the line x = 8:
\\
Region P is the area enclosed by f and g.
Region Q is the area enclosed by f and g and x = 8.
Find the area of region P.
Find the area of region Q.
f \left( x \right) is redefined such that f \left( x \right) = ax and the area of region P is half the area of region Q. Calculate the value of a that makes the statement true.
