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Areas and Volumes

Area Under a Curve

Introduction
The function g(x) = f(t) dt: The fundamental theorem of calculus, part 1 states that if the function f(x) is continuous on a closed interval [a, b], then the function g(x) defined by g(x) = f(t) dt, a ≤ x ≤ b, is continuous on [a, b], and differentiable on (a, b) and g'(x) = f(x).

If the function f(t) ≥ 0, then g(x) ≥ 0. Therefore, g(x) can be interpreted geometrically as the positive value of the area between the curve f(t) and the t–axis from t = a to x.

If the function f(t) ≤ 0, then g(x) ≤ 0. Therefore, g(x) can be interpreted geometrically as the negative value of the area between the curve f(t) and the t–axis from t = a to x.

Approximating the Area Under a Curve

Let y = f(x) be a continuous function defined on the closed interval [a, b]. To find the area of the region that lies under the curve y = f(x) from a to b, where f(x) ≥ 0 for all 'x' in [a, b], divide the interval into 'n' strips of equal width Δx = . These strips divide the interval [a, b] into 'n' sub-intervals, called, [a = x0, x1], [x1, x2], . . ., [xn – 1, xn = b].

In each sub-interval we select some number. Denote the number chosen from the ith sub-interval by ci. Construct an approximating rectangle above the sub-interval [xi – 1, xi] with the height f(ci) and the width Δx. Then the area of this rectangle is: f(ci)Δx.

What we think of intuitively as the area of the region that lies under the curve y = f(x) from a to b is approximated by the sum of the areas of these rectangles, which is: f(c1)Δx + f(c2)Δx + . . . . + f(cn)Δx. This approximation appears to become better and better as the number of strips increases, that is, as n → ∞. Therefore, we define the area of the region that lies under the curve y = f(x) from a to b in the following way.

Definition: The area of the region that lies under the graph of the continuous function 'f' is the limit of the sum of the areas of approximating rectangles: [f(c1)Δx + f(c2)Δx + . . . . + f(ci)Δx + . . . . + f(cn)Δx], where ci is any point in the ith sub-interval [xi – 1, xi].

Left endpoint approximation: If ci is chosen to be the left endpoint of the sub-interval [xi – 1, xi], then ci = xi – 1 and we have: Ln = [f(x0) + f(x1) + . . . . . . . . . . . . + f(xn – 1)]Δx = .

Right endpoint approximation: If ci is chosen to be the right endpoint of the interval [xi – 1, xi], then ci = xi and we have: Rn = [f(x1) + f(x2) + . . + f(xn)]Δx = .

Midpoint rule: If ci is chosen to be the midpoint of the interval [xi – 1, xi], then ci = xi = [xi – 1 + xi] and we have: Mn = [f(x1) + f(x2) + . . . . . . . . . . + f(xn)]Δx = .

Trapezoidal rule: This is another method for approximating the area under a curve, which results from averaging the left endpoint and the right endpoint approximations. Therefore, according to trapezoidal rule, the area of the region that lies under the curve y = f(x) from a to b is: = = .

Simpson's Rule

Trapezoidal rule approximates an integral of a curve using straight line segments. Instead parabolas can also be used. The method is called simpson's rule and the steps are:

  1. Assume 'n' is even.
  2. Divide [a, b] into 'n' sub-intervals of equal length h = (b – a) / n
  3. Approximate the curve y = f(x) ≥ 0 passing the 3 successive points of sub division on by a parabola. Refer below figure.
  4. Compute the area under all the parabolas.
  5. Add the results. We get
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