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Properties of definite integrals practice problems

Linearity, reversing the limits and splitting the interval, from a few known integrals. 10 problems with worked solutions; in 10 of them every equation is proved by a computer algebra system.

Suppose \( \displaystyle \int_0^{6} f(x)\,dx = 4 \), \( \displaystyle \int_0^{3} f(x)\,dx = -2 \), \( \displaystyle \int_0^{6} g(x)\,dx = 9 \) and \( \displaystyle \int_0^{3} g(x)\,dx = 5 \). Find \( \displaystyle \int_{3}^{0} \left(g(x)\right) dx \).

Problem 4.322easy✓ Every equation proved

Suppose \( \displaystyle \int_0^{5} f(x)\,dx = 6 \), \( \displaystyle \int_0^{2} f(x)\,dx = -2 \), \( \displaystyle \int_0^{5} g(x)\,dx = 5 \) and \( \displaystyle \int_0^{2} g(x)\,dx = -1 \). Find \( \displaystyle \int_{5}^{2} \left(f(x) + g(x)\right) dx \).

Problem 4.323easy✓ Every equation proved

Suppose \( \displaystyle \int_0^{5} f(x)\,dx = -1 \), \( \displaystyle \int_0^{2} f(x)\,dx = -6 \), \( \displaystyle \int_0^{5} g(x)\,dx = 4 \) and \( \displaystyle \int_0^{2} g(x)\,dx = 1 \). Find \( \displaystyle \int_{5}^{0} \left(f(x)\right) dx \).

Problem 4.324easy✓ Every equation proved

Suppose \( \displaystyle \int_0^{5} f(x)\,dx = 6 \), \( \displaystyle \int_0^{2} f(x)\,dx = -3 \), \( \displaystyle \int_0^{5} g(x)\,dx = -2 \) and \( \displaystyle \int_0^{2} g(x)\,dx = 6 \). Find \( \displaystyle \int_{2}^{0} \left(f(x) + g(x)\right) dx \).

Problem 4.325easy✓ Every equation proved

Suppose \( \displaystyle \int_0^{3} f(x)\,dx = -4 \), \( \displaystyle \int_0^{2} f(x)\,dx = -2 \), \( \displaystyle \int_0^{3} g(x)\,dx = -3 \) and \( \displaystyle \int_0^{2} g(x)\,dx = 8 \). Find \( \displaystyle \int_{2}^{0} \left(f(x) + 2\right) dx \).

Problem 4.326easy✓ Every equation proved

Suppose \( \displaystyle \int_0^{4} f(x)\,dx = 1 \), \( \displaystyle \int_0^{2} f(x)\,dx = 6 \), \( \displaystyle \int_0^{4} g(x)\,dx = -1 \) and \( \displaystyle \int_0^{2} g(x)\,dx = -1 \). Find \( \displaystyle \int_{2}^{4} \left(g(x)\right) dx \).

Problem 4.327easy✓ Every equation proved

Suppose \( \displaystyle \int_0^{4} f(x)\,dx = 4 \), \( \displaystyle \int_0^{2} f(x)\,dx = 7 \), \( \displaystyle \int_0^{4} g(x)\,dx = 6 \) and \( \displaystyle \int_0^{2} g(x)\,dx = 8 \). Find \( \displaystyle \int_{0}^{4} \left(2 f(x) - 3 g(x)\right) dx \).

Problem 4.328easy✓ Every equation proved

Suppose \( \displaystyle \int_0^{6} f(x)\,dx = 4 \), \( \displaystyle \int_0^{3} f(x)\,dx = 5 \), \( \displaystyle \int_0^{6} g(x)\,dx = 2 \) and \( \displaystyle \int_0^{3} g(x)\,dx = -4 \). Find \( \displaystyle \int_{6}^{3} \left(f(x) + g(x)\right) dx \).

Problem 4.329easy✓ Every equation proved

Suppose \( \displaystyle \int_0^{4} f(x)\,dx = 4 \), \( \displaystyle \int_0^{3} f(x)\,dx = 8 \), \( \displaystyle \int_0^{4} g(x)\,dx = -1 \) and \( \displaystyle \int_0^{3} g(x)\,dx = 2 \). Find \( \displaystyle \int_{0}^{3} \left(- f(x) + 2 g(x)\right) dx \).

Problem 4.330easy✓ Every equation proved

Suppose \( \displaystyle \int_0^{6} f(x)\,dx = -1 \), \( \displaystyle \int_0^{1} f(x)\,dx = 9 \), \( \displaystyle \int_0^{6} g(x)\,dx = -9 \) and \( \displaystyle \int_0^{1} g(x)\,dx = 7 \). Find \( \displaystyle \int_{6}^{1} \left(f(x) + 3\right) dx \).

Problem 4.331easy✓ Every equation proved