Math Snap

STEP 1

1. The problem involves properties of definite integrals.
2. We need to simplify and combine integrals to find the bounds $a$ and $b$.

STEP 2

1. Simplify the given equation by combining integrals.
2. Identify the bounds $a$ and $b$.

STEP 3

Start by simplifying the right-hand side of the equation. We have:
$$\int_{a}^{b} f(x) \, dx = \int_{-2}^{10} f(x) \, dx + \int_{10}^{15} f(x) \, dx - \int_{-2}^{5} f(x) \, dx$$ Combine the integrals on the right-hand side. Notice that:
$$\int_{-2}^{10} f(x) \, dx - \int_{-2}^{5} f(x) \, dx = \int_{5}^{10} f(x) \, dx$$ So the equation becomes:
$$\int_{a}^{b} f(x) \, dx = \int_{5}^{10} f(x) \, dx + \int_{10}^{15} f(x) \, dx$$

STEP 4

Now, combine the two integrals on the right-hand side:
$$\int_{a}^{b} f(x) \, dx = \int_{5}^{15} f(x) \, dx$$

From the equation $\int_{a}^{b} f(x) \, dx = \int_{5}^{15} f(x) \, dx$, we can directly identify the bounds:
$$a = 5$$ $$b = 15$$ The bounds of integration for the first integral are:
$$a = 5$$ $$b = 15$$