12.08 Compound areas and the area between two curves
Find the area bounded between the curve y = \sqrt{x + 5}, the line y = - x - 3 and the x-axis:
Find the area bounded between the curve y = \sqrt{3-x}, the line y=\dfrac{2x+8}{3} and the x-axis:
Consider the functions x + y = 2 and y = x.
Graph the functions on the same number plane.
Hence, find the area bounded between the two lines and the x-axis.
Consider the functions x + y = 6 and y = 2x.
Graph the functions on the same number plane.
Hence, find the area bounded between the two lines and the x-axis.
Consider the functions y = x^{2} and y = \left(x - 2\right)^{2}.
Graph the functions on the same number plane.
State the value of x at which the curves intersect.
Hence, find the area enclosed between the curves and the x-axis.
Consider the functions y = (x-1)^{2} and y = \left(x + 3\right)^{2}.
Graph the functions on the same number plane.
State the value of x at which the curves intersect.
Hence, find the area enclosed between the curves and the x-axis.
Find the area of the region bounded by the following pairs of functions and the x-axis, to two decimal places:
y=2x and y=(x-4)^2.
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 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.
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.
Find the area enclosed between the lines y = 2 x, y = \dfrac{1}{3} x and x = 6:
For each of the following:
Find the values of x at which the line and the curve intersect.
Find the area between the two curves.
y = 2 x and y = x^{2} - 15
y = x and y = \left(x - 5\right)^{3} + 5
y = 4 x - 12 and y = x \left(x - 3\right)^{2}
Consider the functions y = x^{2} - 48 and y = - \left(x - 2\right)^{2} + 4.
Find the values of x at which the two curves intersect.
Find the area enclosed between the two curves.
For each of the following:
Sketch the functions on the same Cartesian plane.
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 - 4\right)^{2} and y = - x
The following diagram shows the curves y = - x^{2} + 4 x - 4 and y = x^{2} - 8 x + 12 meeting at the points \left(2, 0\right) and \left(4, - 4 \right):
Find the area of the shaded region.
The following diagram shows the curves y = x^{2} and y = x^{4}:
Find the area of the shaded region.
Consider the functions y = - 2 x \left(x - 4\right) and y = - x + 4.
Graph the functions on the same number plane.
State the values of x at which the curve and the line intersect.
Hence, find the area enclosed between the curves, correct to one decimal place.
Find the small area enclosed between the curve, the line and the y-axis. Round your answer to one decimal place.
Consider
Graph the functions y = x^{2} and y = 8 - x^{2} on the same Cartesian plane.
State the values of x at which the curves intersect.
Hence, find the area bounded between the curves, correct to one decimal place.
Find the small area bounded between the curves and the x-axis, correct to one decimal place.
Find the exact area between the graph of y = \sqrt{4 - x^{2}} and the lines x = 2 and y = 2.
Find the area between the graph of y = x^{3} - 8 and the lines x = 2 and y = - 8.
Consider the graph of the functions \\ y = x \left(x - 6\right)^{2} and y = x^{2}:
State the values of x at which the curves intersect.
Hence, find the total area bounded between the curves. Round your answer to one decimal place.
Find the area bounded by the curves and the x-axis. Round your answer to one decimal place.