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Applications of Definite Integrals

Average Value of a Function

We define the average value of finitely many numbers as the sum of the numbers divided by number of numbers. How would we define the average value of an arbitrary function f(x) over a closed interval [a, b]? As there are infinitely many numbers to consider, adding them and then dividing by infinity is not an option.

In order to compute the average value of a function f(x) over a closed interval [a, b], divide the interval [a, b] into 'n' equal sub-intervals, each with length Δx = (b – a)/n. Then we choose sample points c1, c2, ....., cn in successive sub-intervals and calculate the average of the numbers f(c1), f(c2), ....., f(cn).

The average value of the numbers f(c1), f(c2), ....., f(cn) is: . Since Δx = (b – a)/n, we can write n = (b – a)/Δx and the average value becomes:

If we let 'n' increase, we would be computing the average value of a large number of closely spaced values. The limiting value is: by the definition of a definite integral. Therefore, we define the average value of f(x) on a closed interval [a, b] as:

  • Ex: Find the average value of the function f(x) = x3 on the interval [1, 3].

    Sol:

Mean value theorem for definite integrals: If the function f(x) is continuous on a closed interval [a, b], then there exists a number 'c' in [a, b] such that: or

Therefore, mean value theorem for definite integrals states that a continuous function on a closed interval always assumes its average value at least once in the interval.

  • Ex: If f(x) is continuous and = 15, show that f(x) takes on the value 5 at least once on the interval [1, 4].

    Sol: By the mean value theorem, there exists a number 'c' in [1, 4] such that f(c) = favg = .
Differential Equations

In general, a differential equation is an equation that involves independent and dependent variables and the derivatives of the dependent variables. For example, when we consider a differential equation y' = xy, it is understood that 'y' is a dependent variable and 'x' is an independent variable. The order of a differential equation is the order of the highest derivative that occurs in an equation. Therefore, the order of a differential equation y' = xy is one.

A function f(x) is called a solution of a differential equation if the equation is satisfied when y = f(x) and its derivatives are substituted into the equation. Thus, f(x) is a solution of an equation y' = xy if f '(x) = xf(x) for all the values of 'x' in some interval.

When we are asked to solve a differential equation we are expected to find all the possible solutions of the equation. But, in general, solving a differential equation is not an easy matter. There is no systematic approach that enables us to solve a differential equations.

  • Ex: Show that every member of the family of functions y = 15 + is a solution of the differential equation y' = 30x – 2xy.
    Sol: Given, y = 15 + = [15 + ] = [15] + []
    = 0 + [– x2]
    = – 2x
    The right side of differential equation becomes: 30x – 2xy = 30x – 2x(15 + )
    = 30x – 30x – 2x
    = – 2x
    ∴ For every value of 'x', the given function is a solution of a differential equation.

When applying differential equations we are usually not as interested in finding the general solution (a family of solutions) as we are in finding a solution that satisfies some additional requirement. In most of the physical problems, we need to find the particular solution (an exact solution) that satisfies a condition of the form y(t0) = y0. This is called an initial condition and the problem of finding a solution of the differential equation that satisfies the initial condition is called an initial value problem.

Slope Fields and Euler's Method

Slope field or Direction field: Suppose we have a first-order differential equation of the form y' = Φ(x, y), where Φ(x, y) is some expression in 'x' and 'y'. The differential equation says that the slope of a solution curve at a point (x, y) on the curve is Φ(x, y). If we draw short line segments with slope Φ(x, y) at several points (x, y), then the result field is called slope field or direction field. These line segments indicate the direction in which a solution curve is moving, so the direction field or slope field helps us visualize the general shape of the curve.

  • Ex: Sketch the direction field for the differential equation y' = 2x + y.

    Sol: We start by computing the slope at several points in the following table:



    Now we draw short line segments with these slopes at these points. The result is the slope field shown in the figure below:

Euler's method: It is a numerical technique to solve first order differential equations of the form: y' = Φ(x, y) with the initial condition y(x0) = y0. It approximates the values of the solution of differential equation at equally spaced numbers: x0, x1 = x0 + h, x2 = x1 + h, ....., where h is the step size.

We start at x-initial value x0 and y-initial value y0, that is, at (x0, y0). The differential equation tells us that the slope at (x0, y0) is y' = Φ(x0, y0), so the tangent line to the solution curve at the initial point is: y – y0 = Φ(x0, y0)(x – x0) ⇒ y = y0 + Φ(x0, y0)(x – x0). For the point x1 = x0 + h, we can compute the y-value of the approximate solution by: y1 = y0 + h · Φ(x0, y0) (see the below figure).

Next, start at the point (x1, y1). The differential equation tells us that the slope at (x1, y1) is y' = Φ(x1, y1), so the tangent line to the solution curve at the point (x1, y1) is: y – y1 = Φ(x1, y1)(x – x1) ⇒ y = y1 + Φ(x1, y1)(x – x1). For the point x2 = x1 + h, we can compute the y-value of the approximate solution by: y2 = y1 + h · Φ(x1, y1). For the point x3 = x2 + h, we can compute the y-value of the approximate solution by: y3 = y2 + h · Φ(x2, y2). For the point x4 = x3 + h, we can compute the y-value of the approximate solution by: y4 = y3 + h · Φ(x3, y3). Continuing on this way, we obtain a sequence of (x, y) values: xn + 1 = xn + h and yn + 1 = yn + h . Φ(xn, yn).

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