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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, $\int_{1}^{2} g(x) , dx = \int_{1}^{4} g(x) , dx - \int_{2}^{4} g(x) , dx = -8 - (-3) = -5$.
Explain why $\int_{a}^{a} f(x) , dx = 0$.
The integral represents the area under the curve. If the upper and lower limits are the same, the 'width' of the area is zero, resulting in zero area.
Explain the concept of reversing the limits of integration.
Reversing the limits changes the direction of integration, thus negating the value of the integral because the area is now considered 'negative'.
Why can a constant be moved outside the integral?
Multiplying a function by a constant scales its area by the same factor. Thus, the constant can be factored out.
Explain the concept of splitting the interval of integration.
The total area under a curve from a to c can be found by summing the area from a to b and then from b to c.
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.