Question

Evaluate the integral. $$ \int_{7}^{12} d x $$

   Evaluate the integral.
$$
\int_{7}^{12} d x
$$

Calculus

Earl W. Swokowski 5th Edition

Chapter 5, Problem 5

Instant Answer

Solved by Expert Gregory Higby

03/15/2022

Step 1

In this case, we are given the integral of dx, which is the same as the integral of 1 dx. The integral of 1 dx is simply x, because the derivative of x is 1. Show more…

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Evaluate the integral. $$ \int_{7}^{12} d x $$

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Key Concepts

Definite Integral

A definite integral represents the accumulation of a quantity, such as the area under a curve, within a specific interval. It computes a numerical value by evaluating the integral from one endpoint to the other, summarizing the net effect (or signed area) of a function over that interval.

Antiderivative

An antiderivative is a function whose derivative results in the original function being integrated. It is used to evaluate definite integrals by applying the process of finding the difference between the antiderivative evaluated at the upper and lower bounds.

Fundamental Theorem of Calculus

The Fundamental Theorem of Calculus connects integration and differentiation by stating that if a function is continuous over a closed interval, then the definite integral over that interval can be determined by subtracting the antiderivative evaluated at the lower limit from the antiderivative evaluated at the upper limit.

Integration of a Constant Function

When integrating a constant function, the process simplifies to multiplying the constant by the width of the interval. For example, integrating a constant value of 1 over an interval results in the length of the interval, since the antiderivative of 1 is just the variable of integration.

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