Definite Integrals

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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) (correct)

φ(x) is the integral of f(x)

φ(x) is the derivative of f(x)

φ(x) is the inverse of f(x)

What is the purpose of the constant of integration when evaluating a definite integral?

It is used to find the lower limit of integration

It is added to the result of the definite integral

It is not used in the evaluation of a definite integral (correct)

It is used to find the upper limit of integration

If ∫f(x) dx = φ(x) + C, what is the value of C when evaluating a definite integral?

It is equal to 0

It can be any value

It is not involved in the evaluation of a definite integral (correct)

It is equal to 1

What is the name of the formula ∫[a, b] f(x) dx = φ(b) - φ(a)?

Newton-Leibniz Formula Signup and view all the answers

What is the interval [a, b] referred to as?

Interval of Integration Signup and view all the answers

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)) Signup and view all the answers

Let f(x) = ∫[x^2 - t^2]dt from 0 to x. What is the value of f'(x)?

2 - x^2 Signup and view all the answers

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 Signup and view all the answers

Let f(x) = ∫[x^2 - t^2]dt from 0 to x. What is the value of x such that f'(x) = 0?

±1 Signup and view all the answers

Let f(x) = ∫[f(t)dt] from 0 to x, and f(x) = x(1 + x). What is the value of f(4)?

16 Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

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 Signup and view all the answers

If A = lim (n → ∞) [(n!)(1/n)], then what is the value of log A?

e - 1 Signup and view all the answers

If ∫[0, 1] f(x)dx = lim (n → ∞) Σf(1/n), then what is the function f(x)?

1/(1 + x) Signup and view all the answers

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) Signup and view all the answers

Why is the constant of integration not needed when evaluating a definite integral?

Because the constant cancels out when evaluating the definite integral Signup and view all the answers

What is the property of a definite integral that allows it to have a unique value?

Uniqueness Signup and view all the answers

What is the name of the formula that states ∫[a, b] f(x)dx = φ(b) - φ(a)?

Newton-Leibnitz Formula Signup and view all the answers

What is the purpose of the limits of integration in a definite integral?

To specify the interval over which the function is integrated Signup and view all the answers

What is the condition required for a function f(x) to be integrable on [a, b]?

f(x) is continuous on [a, b] Signup and view all the answers

If ∫[a, b] f(x) dx = U, then what is the value of ∫[b, a] f(x) dx?

-U Signup and view all the answers

What is the purpose of integration by parts in calculus?

To find the integral of a product of functions Signup and view all the answers

If f(x) is continuous on [a, b], then what is the value of ∫[a, b] f'(x) dx?

f(b) - f(a) Signup and view all the answers

What is the value of lim (n → ∞) [(n!)(1/n)]?

e Signup and view all the answers

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 Signup and view all the answers

Let f(x) = ∫[x^2 - t^2]dt from 0 to x. What is the value of f'(x)?

2x^2 Signup and view all the answers

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) Signup and view all the answers

Let f(x) = ∫[f(t)dt] from 0 to x, and f(x) = x(1 + x). What is the value of f(4)?

16 Signup and view all the answers

What is the purpose of the constant of integration when evaluating a definite integral?

To account for the arbitrary constant in the antiderivative Signup and view all the answers

Definite Integrals

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Questions and Answers

If ∫[a, b] f(x) dx = φ(b) - φ(a), what is the relationship between φ(x) and f(x)?

What is the purpose of the constant of integration when evaluating a definite integral?

If ∫f(x) dx = φ(x) + C, what is the value of C when evaluating a definite integral?

What is the name of the formula ∫[a, b] f(x) dx = φ(b) - φ(a)?

What is the interval [a, b] referred to as?

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?

Let f(x) = ∫[x^2 - t^2]dt from 0 to x. What is the value of f'(x)?

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)?

Let f(x) = ∫[x^2 - t^2]dt from 0 to x. What is the value of x such that f'(x) = 0?

Let f(x) = ∫[f(t)dt] from 0 to x, and f(x) = x(1 + x). What is the value of f(4)?

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?

If ∫[a, b] f(x)dx = lim (h → 0) Σf(a + rh), where h = (b - a)/n, then what is the value of n?

If S_n = 1 + 1/(1 + 1/n) + 1/(2 + 2/n) + ... , then what is the value of lim (n → ∞) S_n?

If A = lim (n → ∞) [(n!)(1/n)], then what is the value of log A?

If ∫[0, 1] f(x)dx = lim (n → ∞) Σf(1/n), then what is the function f(x)?

What is the value of the definite integral ∫[a, b] f(x)dx in terms of the primitive or anti-derivative φ(x)?

Why is the constant of integration not needed when evaluating a definite integral?

What is the property of a definite integral that allows it to have a unique value?

What is the name of the formula that states ∫[a, b] f(x)dx = φ(b) - φ(a)?

What is the purpose of the limits of integration in a definite integral?

What is the condition required for a function f(x) to be integrable on [a, b]?

If ∫[a, b] f(x) dx = U, then what is the value of ∫[b, a] f(x) dx?

What is the purpose of integration by parts in calculus?

If f(x) is continuous on [a, b], then what is the value of ∫[a, b] f'(x) dx?

What is the value of lim (n → ∞) [(n!)(1/n)]?

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?

Let f(x) = ∫[x^2 - t^2]dt from 0 to x. What is the value of f'(x)?

If ∫[a, b] f(x)dx = φ(b) - φ(a), what is the relationship between φ(x) and 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)?

What is the purpose of the constant of integration when evaluating a definite integral?

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