4.08 The fundamental theorem of calculus

Interactive practice questions

The function $f$f has an antiderivative $F$F, and $F\left(3\right)=4$F(3)=4.

Express $\int_3^xf\left(t\right)dt$∫x3f(t)dt in terms of $F$F and $x$x.

Now, calculate $\frac{d}{dx}\int_3^xf\left(t\right)dt$ddx∫x3f(t)dt.

Easy

6min

Consider the function $f\left(t\right)=-4t$f(t)=−4t.

Easy

4min

Consider the function $f\left(t\right)=12t+9$f(t)=12t+9.

Easy

3min

Calculate $\frac{d}{dx}\int_2^x\left(6t^2-8t+3\right)dt$ddx∫x2(6t2−8t+3)dt.

Easy

1min

examine the concept of the signed area function F(x)=∫ {from a to x} f(t)dt

apply the theorem F'(x)=d/dx ( ∫ {from a to x} f(t)dt)=f(x), and illustrate its proof geometrically

use the increments formula: δy≅dy/dx × δx to estimate the change in the dependent variable y resulting from changes in the independent variable x

develop the formula ∫ {from a to b} f(x)dx= F(b)−F(a) and use it to calculate definite integrals