Math / Calculus

QuestionIf $\int_{a}^{b} f(x) d x=\int_{-2}^{10} f(x) d x+\int_{10}^{15} f(x) d x-\int_{-2}^{5} f(x) d x$ what are the bounds of integration for the first integral? $a=$ and $b=$

Studdy Solution

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:
$\int_{a}^{b} f(x) \, dx = \int_{5}^{15} f(x) \, dx$

STEP 5

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:
$a = 5$ $b = 15$

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