Math Problem Statement

If R(x) is an even function and S(x) is an odd function where \int_(-a)^0 R(x)dx=8 and \int_(-a)^0 S(x)dx=-9, find the value of \int_(-a)^a [2(S(x))-R(x)+3]dx

Solution

We are tasked with finding the value of the integral:

I=aa[2S(x)R(x)+3]dxI = \int_{-a}^a \left[ 2S(x) - R(x) + 3 \right] dx

Step 1: Break the integral into simpler parts

We can separate the integral into individual terms:

I=aa2S(x)dxaaR(x)dx+aa3dxI = \int_{-a}^a 2S(x) \, dx - \int_{-a}^a R(x) \, dx + \int_{-a}^a 3 \, dx

We will now analyze each part of this sum separately.

Step 2: Evaluate aa2S(x)dx\int_{-a}^a 2S(x) \, dx

Since S(x)S(x) is an odd function, we know that for any odd function f(x)f(x):

aaf(x)dx=0\int_{-a}^a f(x) \, dx = 0

Thus, because S(x)S(x) is odd:

aaS(x)dx=0\int_{-a}^a S(x) \, dx = 0

Multiplying by 2:

aa2S(x)dx=20=0\int_{-a}^a 2S(x) \, dx = 2 \cdot 0 = 0

Step 3: Evaluate aaR(x)dx\int_{-a}^a R(x) \, dx

Since R(x)R(x) is an even function, we know that for any even function f(x)f(x):

aaf(x)dx=20af(x)dx\int_{-a}^a f(x) \, dx = 2 \int_0^a f(x) \, dx

Thus, for R(x)R(x) being even:

aaR(x)dx=20aR(x)dx\int_{-a}^a R(x) \, dx = 2 \int_0^a R(x) \, dx

From the problem, we know that:

a0R(x)dx=8\int_{-a}^0 R(x) \, dx = 8

By the symmetry of even functions:

0aR(x)dx=8\int_0^a R(x) \, dx = 8

Thus:

aaR(x)dx=2×8=16\int_{-a}^a R(x) \, dx = 2 \times 8 = 16

Step 4: Evaluate aa3dx\int_{-a}^a 3 \, dx

This term is straightforward:

aa3dx=3[x]aa=3(a(a))=32a=6a\int_{-a}^a 3 \, dx = 3 \cdot \left[ x \right]_{-a}^a = 3 \cdot (a - (-a)) = 3 \cdot 2a = 6a

Step 5: Combine all the results

Now that we have evaluated each part, we can combine them:

I=016+6a=6a16I = 0 - 16 + 6a = 6a - 16

Final Answer:

I=6a16\boxed{I = 6a - 16}

Follow-up questions:

  1. How do the properties of odd and even functions simplify integrals?
  2. What are some real-world examples where odd and even functions appear in physics or engineering?
  3. How do the properties of symmetry in integrals apply in multivariable calculus?
  4. Can this method be extended to integrals over other symmetric intervals like [b,b][-b, b] where bab \neq a?
  5. How do transformations like substitution affect integrals of even or odd functions?

Tip: Always check the properties of the functions (odd/even) before attempting integrals; it can drastically simplify the problem!

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Math Problem Analysis

Mathematical Concepts

Integration
Even and Odd Functions
Properties of Symmetry in Integrals

Formulas

∫ from -a to a f(x)dx = 0 for odd functions
∫ from -a to a f(x)dx = 2 ∫ from 0 to a f(x)dx for even functions
∫ from -a to a cdx = 2ac, where c is a constant

Theorems

Properties of Even and Odd Functions in Integration

Suitable Grade Level

Grades 11-12