We are asked to prove the equality:

$\int_a^b f(-x) \, dx = \int_{-a}^{-b} f(x) \, dx.$

Proof of the Equality

We begin with the left-hand side:

$\int_a^b f(-x) \, dx.$

To change the limits of integration, we perform the substitution $u = -x$. Then, $du = -dx$, and the limits change as follows:

• When $x = a$, $u = -a$.
• When $x = b$, $u = -b$.

Thus, the integral becomes:

$\int_a^b f(-x) \, dx = \int_{-a}^{-b} f(u) (-du).$

Since $du = -dx$, we can factor out the negative sign, giving:

$\int_{-a}^{-b} f(u) (-du) = \int_{-b}^{-a} f(u) \, du.$

Since $u$ is a dummy variable (it represents the same thing as $x$), we can replace $u$ with $x$ again:

$\int_{-b}^{-a} f(u) \, du = \int_{-b}^{-a} f(x) \, dx.$

Thus, we have shown that:

$\int_a^b f(-x) \, dx = \int_{-b}^{-a} f(x) \, dx.$

Finally, we can rewrite the right-hand side of the original expression to match this result:

$\int_{-a}^{-b} f(x) \, dx = \int_{-b}^{-a} f(x) \, dx,$

so the equality is proven.

Graphical Interpretation

When $f(x) \geq 0$ and $0 \leq a < b$, the equality can be interpreted as an equality of areas under the curve.

• The integral $\int_a^b f(-x) \, dx$ represents the area under the curve of $f(-x)$, which is a reflection of the function $f(x)$ across the y-axis, evaluated between $a$ and $b$.
• The integral $\int_{-a}^{-b} f(x) \, dx$ represents the area under the curve of $f(x)$, but between the limits $-a$ and $-b$.

The equality shows that the area under the curve on one side of the y-axis is the same as the area on the other side, reflecting the symmetry in the function when transformed by $f(-x)$.

Diagram:

The diagram would consist of two areas:

• One between $a$ and $b$, for the function $f(-x)$, on the positive side of the x-axis.
• One between $-b$ and $-a$, for the function $f(x)$, on the negative side of the x-axis.

The areas under the curves on both sides of the y-axis will be identical if the integral equality holds.

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Do you want any further clarifications or a visual representation of the graph?

Here are some related questions:

1. How does the substitution $u = -x$ work in definite integrals?
2. What is the significance of continuity in proving integral equalities?
3. Can you provide an example of a function $f(x)$ and calculate both integrals?
4. How does symmetry play a role in interpreting these kinds of integrals?
5. What would happen if $f(x)$ were not continuous or had discontinuities?

Tip: When performing a substitution in definite integrals, always adjust the limits of integration accordingly to match the new variable.