Let's address these questions one by one with explanations and solutions where applicable.

Question 1: Using the definition of Riemann integral, evaluate the following:

The question involves evaluating integrals using the definition of the Riemann integral for the given functions:

1. $\int_0^a b^x dx$
2. $\int_0^b e^x dx$
3. $\int_0^{\pi/2} \sin^2 x dx$
4. $\int_0^a \frac{dx}{x}$
5. $\int_0^a (1 + x) dx$

These integrals can be computed by standard techniques in calculus, such as substitution or known integral formulas.

Question 2: Determine which of the following functions are integrable over the indicated intervals.

This question involves checking the integrability of functions defined piecewise, where many of them involve rational and irrational domains. A function defined on an interval $[a, b]$ is integrable if it is bounded and has only a finite number of discontinuities in that interval.

Approach:

1. For each function, analyze whether it is bounded on the given interval.
2. Check if it has a countable number of discontinuities (as functions with uncountable discontinuities are generally not Riemann integrable).

Question 3: Derive the Mean Value Theorem for integrals by applying Lagrange's Mean Value Theorem to the function $F(x) = \int_a^x f(t) dt$.

To apply the Mean Value Theorem, rewrite $F(x)$ in terms of the Mean Value Theorem for integrals:

1. State the conditions for $f$ under which the Mean Value Theorem can be applied (continuity on $[a, b]$).
2. Conclude that there exists a $c \in (a, b)$ such that $F(b) - F(a) = f(c)(b - a)$.

Question 4: Verify the Mean Value Theorem for the following definite integrals.

1. $\int_0^{\pi} \sin x dx$
2. $\int_0^1 e^x dx$
3. $\int_0^4 x^2 dx$

For each integral:

1. Calculate the definite integral value.
2. Identify the point $c$ that satisfies the theorem.

Question 5: Find $F'(x)$ for the given integral functions $F(x)$.

1. Use the Fundamental Theorem of Calculus to differentiate each given function $F(x)$ if it’s expressed as an integral with an upper bound depending on $x$.

Question 6: Prove that $\frac{dy}{dx} = y$ when $x$ and $y$ are related by $x = \int_0^y \frac{dt}{\sqrt{1 + t^2}}$.

This problem requires differentiating the integral relationship given between $x$ and $y$ using implicit differentiation and the chain rule.

Question 7: Prove that $\int_a^b f(x)g(x) dx = f(c) \int_a^b g(x) dx$ for some $c \in [a, b]$, given that $f$ and $g$ are continuous and $g(x)$ does not change sign on $[a, b]$.

For this proof:

1. Use the Mean Value Theorem for integrals, given the continuity of $f$ and $g$ and the constant sign of $g(x)$ over the interval.

Would you like detailed solutions to any specific questions from this list?

Related Questions

1. How can we determine if a function is integrable over a certain interval?
2. What conditions are necessary to apply the Mean Value Theorem in integral calculus?
3. How do we handle piecewise functions when determining integrability?
4. What is the significance of continuity in proving the Mean Value Theorem?
5. How does the Fundamental Theorem of Calculus relate to finding derivatives of integral functions?

Tip

When dealing with Riemann integrals of piecewise functions, always check the count and type of discontinuities, as this affects their integrability over the interval.