30 Questions
If ∫[a, b] f(x) dx = φ(b) - φ(a), what is the relationship between φ(x) and f(x)?
φ(x) is the anti-derivative of f(x)
What is the purpose of the constant of integration when evaluating a definite integral?
It is not used in the evaluation of a definite integral
If ∫f(x) dx = φ(x) + C, what is the value of C when evaluating a definite integral?
It is not involved in the evaluation of a definite integral
What is the name of the formula ∫[a, b] f(x) dx = φ(b) - φ(a)?
Newton-Leibniz Formula
What is the interval [a, b] referred to as?
Interval of Integration
If f(x) is continuous and u(x), v(x) are differentiable functions in the interval [a, b], then what is the value of the integral ∫[v(x)f'(t) - u(x)f'(t)]dt?
f(v(x)) - f(u(x))
Let f(x) = ∫[x^2 - t^2]dt from 0 to x. What is the value of f'(x)?
2 - x^2
If f(x) is continuous and differentiable in the interval [a, b], and ∫f(t)dt from a to b equals x^2, then what is the value of f'(x)?
2x
Let f(x) = ∫[x^2 - t^2]dt from 0 to x. What is the value of x such that f'(x) = 0?
±1
Let f(x) = ∫[f(t)dt] from 0 to x, and f(x) = x(1 + x). What is the value of f(4)?
16
If f(x) is a continuous function on [a, b] and f1(x) and f2(x) are two continuous functions on [a, b] such that ∫[a, b] f1(x)dx ≤ ∫[a, b] f(x)dx ≤ ∫[a, b] f2(x)dx, then which of the following is true?
U ∫[a, b] f1(x)dx ≤ U ∫[a, b] f(x)dx ≤ U ∫[a, b] f2(x)dx
If ∫[a, b] f(x)dx = lim (h → 0) Σf(a + rh), where h = (b - a)/n, then what is the value of n?
n = (b - a)/h
If S_n = 1 + 1/(1 + 1/n) + 1/(2 + 2/n) + ... , then what is the value of lim (n → ∞) S_n?
2 log 2
If A = lim (n → ∞) [(n!)(1/n)], then what is the value of log A?
e - 1
If ∫[0, 1] f(x)dx = lim (n → ∞) Σf(1/n), then what is the function f(x)?
1/(1 + x)
What is the value of the definite integral ∫[a, b] f(x)dx in terms of the primitive or anti-derivative φ(x)?
φ(b) - φ(a)
Why is the constant of integration not needed when evaluating a definite integral?
Because the constant cancels out when evaluating the definite integral
What is the property of a definite integral that allows it to have a unique value?
Uniqueness
What is the name of the formula that states ∫[a, b] f(x)dx = φ(b) - φ(a)?
Newton-Leibnitz Formula
What is the purpose of the limits of integration in a definite integral?
To specify the interval over which the function is integrated
What is the condition required for a function f(x) to be integrable on [a, b]?
f(x) is continuous on [a, b]
If ∫[a, b] f(x) dx = U, then what is the value of ∫[b, a] f(x) dx?
-U
What is the purpose of integration by parts in calculus?
To find the integral of a product of functions
If f(x) is continuous on [a, b], then what is the value of ∫[a, b] f'(x) dx?
f(b) - f(a)
What is the value of lim (n → ∞) [(n!)(1/n)]?
e
If f(x) is a continuous and differentiable function on [a, b] and v(x) and u(x) are differentiable functions on [a, b], then what is the value of the integral ∫[v(x)f'(t) + u(x)f'(t)]dt?
v(x)f(x) - u(x)f(x) + C
Let f(x) = ∫[x^2 - t^2]dt from 0 to x. What is the value of f'(x)?
2x^2
If ∫[a, b] f(x)dx = φ(b) - φ(a), what is the relationship between φ(x) and f(x)?
φ(x) is the antiderivative of f(x)
Let f(x) = ∫[f(t)dt] from 0 to x, and f(x) = x(1 + x). What is the value of f(4)?
16
What is the purpose of the constant of integration when evaluating a definite integral?
To account for the arbitrary constant in the antiderivative
Learn about the definition of definite integrals, its notation, and the Newton-Leibniz formula. Understand the concept of limits of integration and the interval of integration.
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