Evaluate each definite integral to two decimal places. ∫0^30 500 e^0.02 t e^0.09(30-t) d t

Name: Problem 13, Chapter 6: Calculus for Business, Economics, Life Sciences, and Social Sciences
Uploaded: 2021-04-12T22:55:31-08:00
Duration: 2 min 49 s
Channel: Angela Guo
Description: Evaluate each definite integral to two decimal places. ∫0^30 500 e^0.02 t e^0.09(30-t) d t

step 1 We wish to evaluate this definite integral here. So let's simplify this expression here first. S

Evaluate each definite integral to two decimal places. ∫0^30...

Key Concepts

Constant Factor Rule

The constant factor rule in integration permits any constant multiplier of a function to be factored out of the integral. This simplification makes the integration process more straightforward by isolating the variable-dependent part of the integrand.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus bridges the concepts of differentiation and integration. It states that if a function is continuous over an interval, then its definite integral can be computed using its antiderivative, by evaluating the antiderivative at the endpoints of the interval and subtracting these values.

Integration Techniques for Exponential Functions

Integrating exponential functions typically involves recognizing that the derivative of e^(kt) is k e^(kt) and therefore applying the formula ? e^(kt) dt = (1/k)e^(kt) + C, where k is a constant. This technique is essential for handling integrals where the exponent is a linear function of the variable.

Exponential Functions

Exponential functions are mathematical functions in which the variable appears in the exponent, such as e^(kt). They are characterized by rapid growth or decay and are frequently encountered in applications across natural sciences, engineering, and finance.

Definite Integration

Definite integration involves calculating the net area under a curve within specified bounds. This concept is central to integral calculus and is applied by finding the antiderivative of the integrand and then evaluating it at the upper and lower limits, subtracting the latter from the former.

Properties of Exponents

The properties of exponents allow for the simplification of expressions involving exponential functions. One key property is that the product of exponential functions with the same base can be combined by adding their exponents, i.e., e^(a) * e^(b) = e^(a+b), which is crucial in simplifying integrands before integration.

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