Question

Multiple Choice If \(\int_{3}^{5} f(x-a) d x=7\) where \(a\) is a constant then \(\int_{3-a}^{5-a} f(x) d x=\) (A) \(7+a\) (B) 7 (C) \(7-a\) (D) \(a-7 \quad(\mathbf{E})-7\)

Short answer

The correct answer is (B) 7

Step-by-step solution

Title: Understand the Problem

We have been given that \(\int_{3}^{5} f(x-a) d x = 7\), where \(a\) is a constant. We are asked to find the value of \(\int_{3-a}^{5-a} f(x) d x\). This involves applying the substitution rule.

Title: Applying Substitution

We can apply the substitution rule, which states that if we have an integral of \(f(x)\) and we replace \(x\) with \((x-a)\) or \((x+a)\), the value of the integral doesn't change. Hence, let \(u = x-a\). Substituting it back, we get \(x = u+a\). Now, substituting in the given integral, it becomes \(\int_{3-a}^{5-a} f(u) du = 7\). The limits of the integral also change accordingly.

Title: Answer

So, we have \(\int_{3-a}^{5-a} f(x) dx = 7\). This means, the value of the integral after shifting by \(a\) units is the same. Hence, the answer is 7.

Key concepts

Integration

Integration is a fundamental concept in calculus, often considered the inverse operation to differentiation. It allows us to find the total accumulation of a quantity, such as area under a curve or the total distance traveled given a speed function over time.

When we use a definite integral, we are specifically looking for this accumulated value over a particular interval. This is represented by the notation \( \int_{a}^{b} f(x) \, dx \), where \( a \) and \( b \) are the lower and upper limits of integration, respectively, and \( f(x) \) is the function we’re integrating. The process involves finding a function, known as the antiderivative, whose derivative is the given function \( f(x) \) and then evaluating this antiderivative at both limits \( a \) and \( b \) to find the difference.

Substitution Rule in Integration

The substitution rule, often referred to as u-substitution, is a technique used to simplify the integration process. This rule is especially helpful when dealing with integrals of composite functions where direct integration is not straightforward.

The essence of this method involves identifying a part of the integrand that can be replaced with a new variable \( u \) to simplify the expression. One then finds the derivative of \( u \) with respect to \( x \) to obtain \( du \), which can replace \( dx \) in the integral. After substitution, we gain an integral in terms of \( u \), which should be easier to integrate. After finding the antiderivative, we substitute back the original variables to get the solution in the original terms.

Definite Integral

A definite integral has clear start and end points, indicating the accumulated value of a function over a certain interval. Unlike an indefinite integral, which represents a family of functions (antiderivatives) without specific bounds, a definite integral gives a numerical answer that can often have real-world interpretations like area or volume.

To calculate a definite integral, we find the antiderivative of the function and evaluate it at the upper limit, then subtract the value of the antiderivative at the lower limit. This operation is known as the Fundamental Theorem of Calculus and is symbolically represented as \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \), where \( F \) is the antiderivative of \( f \) evaluated at the bounds \( a \) and \( b \).

Integral with Shifted Limits

An integral with shifted limits illustrates an interesting property of definite integrals: shifting a function horizontally does not change the definite integral's value, provided the limits of integration are shifted by the same amount.

For example, in the exercise given, a horizontal shift is applied to \( f(x) \) to obtain \( f(x - a) \) while the limits of integration are adjusted from \( [3, 5] \) to \( [3 - a, 5 - a] \). This shift preserves the area under the curve between the new limits, so the value of the definite integral remains the same. This concept is used when employing the substitution rule, where variables are changed to simplify the integration process. Understanding this property can simplify the calculation of integrals involving transformed functions.