Business Calculus Improper Integrals, Differential Equations
5.3 Improper Integrals Consider an integral to represent the area under the curve shown from x = 0 to x = ∞ : Question: could such an area be finite?
Recall the Basic Area Integral = To determine the area (whether finite or infinite), we set up an integral. Since the limits of integration in our example are not both finite numbers, we call this an improper integral. For our example, area = Since ∞ is not a number, we cannot simply use the first fundamental theorem of calculus. We must rewrite the integral to become a definite integral.
Create a definite integral by arbitrarily stopping the right side of the interval at some finite, but not designated value, b. This creates a definite integral, which can be evaluated. Then, to get the full area from 0 to ∞, we allow b to ‘slide’ to ∞ using a limit.
For our example: Area = = To evaluate this area, we work out the definite integral first, copying the limit at each step. When the definite integral is evaluated, we take the limit as b approaches infinity. If this limit yields a finite number, then the area is finite, and we say the improper integral converges. If this limit gives an infinite result, then the area is infinite, and we say the integral diverges.
5.7 Differential Equations A differential equation is an equation which involves a derivative. examples of differential equations: To solve a differential equation means to find the original function whose derivative appears in the equation.
Steps to solve a differential equation: I.Isolate the derivative, if necessary. II.If the derivative is equal to a formula involving the input variable only, then the original function (solution) is just the antiderivative. III.If the derivative is equal to a formula involving the output variable, then the variables must be separated. This is called the separation of variables technique. IV.If additional information is given, solve for the constant of integration. Note: variables can only be separated using multiplication or division, never addition or subtraction.
Separation of Variables Example: solve Note: the derivative is isolated, and is in the correct form. Multiply both sides by dx. Separation means leave any x variables with dx, and move any y variables to the dy side of the equation. Divide both sides by y.
Next, integrate both sides of the equation. Be careful to notice the variable of the integral in each case. If possible, solve for the output variable, in this case y.