5 Integrals and Their Applications Online Quiz Questions

What does the definite integral $\int_a^b f(x)\,dx$ represent when $f$ changes sign on $[a,b]$?

True or false: An indefinite integral must include an arbitrary constant because different antiderivatives of the same function can differ by a constant.

For a partition of $[a,b]$ into $n$ equal subintervals, which sample point produces a right-endpoint Riemann sum?

If $A(x)=\int_a^x f(t)\,dt$ and $f$ is continuous, what is $A'(x)$?

Select all statements that are valid properties of definite integrals.

For the integral $\int 2x(x^2+4)^5\,dx$, which substitution most directly simplifies the integrand?

True or false: When substitution is used on a definite integral, the limits should be changed to the corresponding values of the new variable if the calculation is to remain entirely in that variable.

Evaluate the definite integral $\int_1^3(2x+1)\,dx$. Enter the exact numerical value.

Select all conclusions that follow from the positivity, comparison, and bound properties of definite integrals.

Explain both parts of the Fundamental Theorem of Calculus and describe how each part connects differentiation and integration.

For which exponent does the power-rule formula $\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$ not apply?

In $\int (3x^2+1)\,dx$, which part is the integrand?

True or false: For an integrable function, $\int_b^a f(x)\,dx=-\int_a^b f(x)\,dx$.

Which expression is an antiderivative of $6x^2-4x+5$?

Suppose $f(x)<0$ for every $x$ in $[a,b]$, where $a<b$. What can be concluded about $\int_a^b f(x)\,dx$?

In $A(x)=\int_a^x f(t)\,dt$, what term describes the role of $t$ inside the integral?

Which sample-point formula produces a midpoint Riemann sum for a partition of $[a,b]$ into equal subintervals of width $\Delta x$?

Evaluate $\int_0^2(3x^2+2)\,dx$. Enter the exact value.