Split a region at each axis crossing and add its areas.
Find every root inside the interval and split the integral there.
Work out the signed integral on each subinterval.
Add their magnitudes for total area. Taking the magnitude of the combined integral can lose area.
Example 1
Find the total shaded area.
The graph crosses the axis at .An antiderivative of the function.Integrate from -2 to 0, below the axis.Integrate from 0 to 2, above the axis.Add the two positive areas.
Example 2
Find the total shaded area.
The graph crosses the axis at .An antiderivative of the function.Integrate from -2 to -1, below the axis.Integrate from -1 to 3, above the axis.Add the two positive areas.
Your turn
Question 1
Find the total shaded area.
Check answer
The graph crosses the axis at .An antiderivative of the function.Integrate from -2 to -1, below the axis.Integrate from -1 to 0, above the axis.Add the two positive areas.
Area is measured in square units.
Question 2
Find the total shaded area.
Check answer
The graph crosses the axis at .An antiderivative of the function.Integrate from -2 to 1, below the axis.Integrate from 1 to 4, above the axis.Add the two positive areas.
Area is measured in square units.
Question 3
Find the total shaded area.
Check answer
The graph crosses the axis at .An antiderivative of the function.Integrate from -2 to 0, below the axis.Integrate from 0 to 5, above the axis.Add the two positive areas.
Area is measured in square units.
Question 4
Find the total shaded area.
Check answer
The graph crosses the axis at .An antiderivative of the function.Integrate from -1 to 2, below the axis.Integrate from 2 to 3, above the axis.Add the two positive areas.
Area is measured in square units.
Question 5
Find the total shaded area.
Check answer
The graph crosses the axis at .An antiderivative of the function.Integrate from -1 to 0, below the axis.Integrate from 0 to 2, above the axis.Add the two positive areas.
Area is measured in square units.
Find the total shaded area.
Answer
The graph crosses the axis at .An antiderivative of the function.Integrate from -2 to -1, below the axis.Integrate from -1 to 0, above the axis.Add the two positive areas.
Area is measured in square units.
Find the total shaded area.
Answer
The graph crosses the axis at .An antiderivative of the function.Integrate from -2 to 1, below the axis.Integrate from 1 to 4, above the axis.Add the two positive areas.
Area is measured in square units.
Find the total shaded area.
Answer
The graph crosses the axis at .An antiderivative of the function.Integrate from -2 to 0, below the axis.Integrate from 0 to 5, above the axis.Add the two positive areas.
Area is measured in square units.
Find the total shaded area.
Answer
The graph crosses the axis at .An antiderivative of the function.Integrate from -1 to 2, below the axis.Integrate from 2 to 3, above the axis.Add the two positive areas.
Area is measured in square units.
Find the total shaded area.
Answer
The graph crosses the axis at .An antiderivative of the function.Integrate from -1 to 0, below the axis.Integrate from 0 to 2, above the axis.Add the two positive areas.