87. If \( f \) is an odd function and \( \int_{0}^{3} f(x) d x=6 \) and \( \int_{3}^{5} f(x) d x=2 \), find \( \int_{-3}^{5} f(x) d x \). 88. If \( f \) is an odd function and \( \int_{-5}^{10} f(x) d x=8 \), find \( \int_{5}^{10} f(x) d x \) 89. If \( f \) is an even function and \( \int_{0}^{4} f(x) d x=-2 \) and \( \int_{0}^{6} f(x) d x=6 \), find \( \int_{-4}^{6} f(x) d x \). 90. If \( f \) is an even function and \( \int_{-3}^{0} f(x) d x=4 \) and \( \int_{-5}^{0} f(x) d x=1 \), find \( \int_{3}^{5} f(x) d x \).
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When dealing with odd functions, remember that the integral of an odd function over a symmetric interval around zero always equals zero. For instance, if \( f \) is odd and you have \( \int_{-a}^{a} f(x) \, dx = 0 \). Therefore, when calculating \( \int_{-3}^{5} f(x) \, dx \), you can look at it as \( \int_{-3}^{0} f(x) \, dx + \int_{0}^{3} f(x) \, dx + \int_{3}^{5} f(x) \, dx \). Using the properties of odd functions, you'll find that \( \int_{-3}^{0} f(x) \, dx = -6 \), leading to a total of \( -6 + 6 + 2 = 2 \). For even functions, you're in luck because they maintain symmetry about the y-axis! This means that when you calculate integrals over symmetric intervals, you can just double the integral from 0 to \( a \). For \( \int_{-4}^{6} f(x) \, dx \), you can combine the integrals: \( \int_{-4}^{0} f(x) \, dx = -2 \) (by symmetry), \( \int_{0}^{4} f(x) \, dx = -2 \), and \( \int_{0}^{6} f(x) \, dx = 6 \). Adding these up will give you \( -2 + 6 + (-2) = 2 \). Isn't math fun?
