Further integration & trapezium rule

The problems of finding gradients of curves and of finding areas under curves may not at sight, seem connected, but calculus solves both these problems. As we have seen, differentiating gives gradient of curves. In this chapter we will show how integration gives the area under the curve.

This powerful tool enables, for example, the area of an aeroplane's wing to be calculated precisely and is widely used in numerous fields such as economics and engineering.

Evaluate a definite integral

Properties of Definite Integrals

The following properties are helpful when calculating definite integrals.

Example of definite integral:

Area Under a Curve by Integration

When the area is above the x axis, the area is positive. <br />When the area is below the x axis, the area is negative.<br />

f(x) < 0 for some interval in [a,b] b  a f ( x)dx  Area of R1  Area of R2  Area of R3 y ...

Area Between Two CurvesLet f and g be continuous functions such that f(x)>g(x)on the interval [a,b]. Then the area of the ...

Fundamental Theorem of Calculus• If f is continuous on [a,b], then the definite integral is b a ...

Evaluating Definite Integrals bTo find  f ( x)dx aFirst find the indefinite Integral  f ( x)dx F ( x) ...

Example:

Trapezium Rule

Definite integrals ( those with limits) have been evaluated exactly, in particular to find the area 'under the curve'. There are many functions which can not be integrated( or can not be integrated by simple methods). In these situations, methods of approximating the integral are needed. One method is to estimate the integral by finding an approximation to the area.
Formula: