The definite integral gives the signed area under a curve. Areas above the x-axis give a positive definite integral. Areas below the x-axis gives a negative definite integral.
We normally think areas are positive, therefore it is important to separate positive and negative integration and use the magnitudes (instead of the signs). We use the absolute value of the definite integral.
Remember to sketch the area!!!
1. Find the area enclosed by the curve and the x-axis.
So the area is 4.5 square units.
2. Find the area enclosed between the curve , the x-axis, and the lines x=-1 and x=2.
Area Enclosed by the y-axis
To find the area between a curve and the y-axis, we change the subject of the equation to x. Therefore:
So the integral is as follows:
1. Find the area enclosed by the curve , the y-axis and the lines y=1 and y=3.
Notice that the function x is now with variable y. Therefore, the curve will become a conic and in this case, a parabolic curve that is rotated 90 degrees to the right (positive).
Therefore, the area is given by:
2. Find the area enclosed between the curve , the y-axis, and the lines y=0 and y=3.
Sums and Differences of Areas
In calculus, you could also look for areas between 2 bounded regions. The general method is taking away the primitive values of the upper function (above) by the lower function (below). The resultant value will be the area of the desired region.
1. Find the area enclosed between the curve , the y-axis, and the lines y=0 and y=4 in the first quadrant.
Sometimes, you need to find the intersection of the two functions first in order to determine the boundaries.
2. Find the point of intersection of and the line . Hence, find the area enclosed between the two curves.
To find the boundaries we need to use simultaneous equations to solve, then find their primitive functions and insert the values in.