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If $\int_{1}^{5} f(x) , dx = 3$ and $\int_{5}^{10} f(x) , dx = 5$ , find $\int_{1}^{10} f(x) , dx$ .

Use the property $\int_{a}^{b} f(x) , dx + \int_{b}^{c} f(x) , dx = \int_{a}^{c} f(x) , dx$ . Thus, $\int_{1}^{10} f(x) , dx = 3 + 5 = 8$ .

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If $\int_{1}^{5} f(x) , dx = 3$ and $\int_{5}^{10} f(x) , dx = 5$ , find $\int_{1}^{10} f(x) , dx$ .

Use the property $\int_{a}^{b} f(x) , dx + \int_{b}^{c} f(x) , dx = \int_{a}^{c} f(x) , dx$ . Thus, $\int_{1}^{10} f(x) , dx = 3 + 5 = 8$ .

If $\int_{1}^{10} g(x) , dx = 12$ and $\int_{6}^{10} g(x) , dx = -7$ , find $\int_{1}^{6} g(x) , dx$ .

Use the property $\int_{a}^{c} f(x) , dx - \int_{b}^{c} f(x) , dx = \int_{a}^{b} f(x) , dx$ . Thus, $\int_{1}^{6} g(x) , dx = 12 - (-7) = 19$ .

If $\int_{1}^{10} f(x) , dx = 15$ and $\int_{10}^{6} f(x) , dx = 12$ , find $\int_{1}^{6} f(x) , dx$ .

First, reverse the limits: $\int_{6}^{10} f(x) , dx = -12$ . Then, $\int_{1}^{6} f(x) , dx = \int_{1}^{10} f(x) , dx - \int_{6}^{10} f(x) , dx = 15 - (-12) = 27$ .

If $\int_{1}^{19} h(x) , dx = 17$ , $\int_{6}^{19} h(x) , dx = 2$ , and $\int_{4}^{6} h(x) , dx = -3$ , find $\int_{1}^{4} h(x) , dx$ .

$\int_{1}^{4} h(x) , dx = \int_{1}^{19} h(x) , dx - \int_{6}^{19} h(x) , dx - \int_{4}^{6} h(x) , dx = 17 - 2 - (-3) = 18$ .

If $\int_{1}^{8} f(x) , dx = -8$ and $\int_{8}^{30} f(x) , dx = 200$ , find $\int_{1}^{30} f(x) , dx$ .

$\int_{1}^{30} f(x) , dx = \int_{1}^{8} f(x) , dx + \int_{8}^{30} f(x) , dx = -8 + 200 = 192$ .

If $\int_{1}^{4} g(x) , dx = -8$ and $\int_{4}^{2} g(x) , dx = 3$ , find $\int_{1}^{2} g(x) , dx$ .

First, reverse the limits: $\int_{2}^{4} g(x) , dx = -3$ . Then, $\int_{1}^{2} g(x) , dx = \int_{1}^{4} g(x) , dx - \int_{2}^{4} g(x) , dx = -8 - (-3) = -5$ .

What is the Zero Rule for definite integrals?

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

What is the formula for reversing limits of integration?

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

How do you handle a constant multiple inside a definite integral?

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

How do you integrate a sum or difference of functions?

$\int_{a}^{b} [f(x) pm g(x)] , dx = \int_{a}^{b} f(x) , dx pm \int_{a}^{b} g(x) , dx$

State the formula for splitting an interval of integration.

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

What is a definite integral?

The value obtained by integrating a function over a specific interval [a, b], representing the area under the curve.

Define the upper limit of integration.

The value 'b' in the definite integral $\int_{a}^{b} f(x) , dx$ , representing the upper bound of the interval.

Define the lower limit of integration.

The value 'a' in the definite integral $\int_{a}^{b} f(x) , dx$ , representing the lower bound of the interval.

What is the integrand?

The function $f(x)$ being integrated in a definite integral.