Application of Integrals Mind Map

The area bounded by the curve y = f(x), the x-axis, and the vertical lines x = a and x = b is found by adding thin vertical strips from x = a to x = b. If f(x) is non-negative on [a, b], the area is ∫_a^b f(x) dx.

Before integrating, mark the limits on the x-axis and check whether the curve stays on one side of the x-axis. This avoids losing marks due to signed-area confusion.

Area bounded by standard curves is found by first sketching the known curve shape, identifying the required bounded region, and then setting a definite integral with correct limits. Common curves include circles, parabolas, ellipses and straight lines.

For standard curves, write the curve in the form needed by the strip direction. Use vertical strips for y as a function of x and horizontal strips for x as a function of y.

The area between two curves y = f(x) and y = g(x) from x = a to x = b is ∫_a^b [upper curve - lower curve] dx, provided the same curve stays above on the whole interval.

After finding intersection points, test one x-value between the limits to decide the top curve. This small check prevents the most common sign error.