Here are practice questions on the two fundamental theorems of Calculus. These exercises are designed to help you better understand and apply these theorems. Reviewing the theorems beforehand may be helpful.
True or False: The second fundamental theorem of calculus states that if \[ F(x) = \int_{a}^{x} f(t) \, dt \] then \[ F'(x) = f(x). \]
Answer: True.
True or False: If \[ F(x) = \int_{-2}^{3x} \sin(t) \, dt \] then the second fundamental theorem of calculus can be used to evaluate \(F'(x)\) as \[ F'(x) = \sin(3x). \]
Answer: False. Note that the upper limit in the integral is \(3x\), not \(x\), so the integral has the form \[ F(x) = \int_{-2}^{u(x)} f(t) \, dt. \] Using the chain rule, we get \[ F'(x) = \frac{dF}{du} \cdot \frac{du}{dx} = 3 \sin(3x). \]
True or False: Using the first fundamental theorem of calculus \[ \int_a^b f(x) \, dx = F(b) - F(a), \] we can evaluate \[ \int_{-1}^{1} \frac{1}{x^2} \, dx = -2. \]
Answer: False. The interval of integration \([-1, 1]\) contains 0, where \( \frac{1}{x^2} \) is discontinuous. Therefore, the theorem cannot be applied in this case.
Additional questions with solutions and tutorials and problems.