Given that [tex]$\int_1^5 f(x) d x=8$[/tex], evaluate [tex]$\int_3^5 f(x) d x+\int_3^1\left[\frac{1}{3 x^2}-f(x)\right] d x$[/tex].
Answer (1)
Use the property ∫ a b f ( x ) d x = − ∫ b a f ( x ) d x to rewrite the second integral.
Distribute the integral and combine the integrals involving f ( x ) to get ∫ 1 5 f ( x ) d x − ∫ 1 3 3 x 2 1 d x .
Substitute the given value ∫ 1 5 f ( x ) d x = 8 and evaluate the integral ∫ 1 3 3 x 2 1 d x = 9 2 .
Calculate the final result: 8 − 9 2 = 9 70 .
Explanation
Problem Setup We are given that ∫ 1 5 f ( x ) d x = 8 , and we want to evaluate ∫ 3 5 f ( x ) d x + ∫ 3 1 [ 3 x 2 1 − f ( x ) ] d x .
Reversing the Limits of Integration First, we can use the property of definite integrals that ∫ a b g ( x ) d x = − ∫ b a g ( x ) d x . Applying this to the second integral, we have ∫ 3 1 [ 3 x 2 1 − f ( x ) ] d x = − ∫ 1 3 [ 3 x 2 1 − f ( x ) ] d x
Distributing the Integral Next, we distribute the integral: − ∫ 1 3 [ 3 x 2 1 − f ( x ) ] d x = − ∫ 1 3 3 x 2 1 d x + ∫ 1 3 f ( x ) d x
Substituting Back Now, we substitute this back into the original expression: ∫ 3 5 f ( x ) d x + ∫ 3 1 [ 3 x 2 1 − f ( x ) ] d x = ∫ 3 5 f ( x ) d x − ∫ 1 3 3 x 2 1 d x + ∫ 1 3 f ( x ) d x
Combining Integrals We can combine the integrals involving f ( x ) :
∫ 3 5 f ( x ) d x + ∫ 1 3 f ( x ) d x = ∫ 1 5 f ( x ) d x
Using the Given Information We are given that ∫ 1 5 f ( x ) d x = 8 , so we have: ∫ 3 5 f ( x ) d x + ∫ 3 1 [ 3 x 2 1 − f ( x ) ] d x = 8 − ∫ 1 3 3 x 2 1 d x
Evaluating the Integral Now, we evaluate the integral ∫ 1 3 3 x 2 1 d x . We can rewrite this as 3 1 ∫ 1 3 x − 2 d x . The antiderivative of x − 2 is − x − 1 = − x 1 . Therefore, ∫ 1 3 3 x 2 1 d x = 3 1 [ − x 1 ] 1 3 = 3 1 ( − 3 1 − ( − 1 ) ) = 3 1 ( 1 − 3 1 ) = 3 1 ( 3 2 ) = 9 2
Final Calculation Finally, we substitute this value back into the expression: 8 − 9 2 = 9 72 − 9 2 = 9 70
Final Answer Therefore, the value of the given expression is 9 70 .
Examples
Definite integrals are used extensively in physics to calculate quantities such as displacement, velocity, and acceleration. For example, if f ( x ) represents the velocity of an object at time x , then ∫ a b f ( x ) d x gives the displacement of the object between times a and b . Similarly, in engineering, definite integrals are used to calculate areas, volumes, and moments of inertia, which are crucial for designing structures and machines. Understanding how to manipulate and evaluate definite integrals is essential for solving real-world problems in these fields.
