Question

State the Fundamental Theorem of Calculus

(a) in its original form,

(b) in its alternative form, and

(c) by using an indefinite integral and evaluation notation.

Short answer

(a) The fundamental Theorem of calculus in its original form is ${\int }_a^{b}f\left(x\right).dx=F\left(b\right)-F\left(a\right)$.

(b) The fundamental Theorem of calculus in its alternative form is${\left[F\right(x\left)\right]}_a^b=F\left(b\right)-F\left(a\right)$.

(c) The fundamental Theorem of calculus in its by using an indefinite integral and evaluation notation is ${\int }_a^{b}f\left(x\right).dx={[\int f(x).dx]}_a^b$.

Step-by-step solution

Part (a) Step 1. Given Information.

The fundamental theorem of calculus.

Part (a) Step 2. Original form.

If f is continuous on [a, b] and F is any antiderivative of f, then

${\int }_a^{b}f\left(x\right).dx=F\left(b\right)-F\left(a\right)$

Part (b) Step 1. Alternative form.

For any function F on an interval [a, b], the difference F(b)−F(a) will be called the evaluation of F(x) on[a, b] and will be denoted by ${\left[F\right(x\left)\right]}_a^b=F\left(b\right)-F\left(a\right)$.

Part (c) Step 1. Evaluation Notation.

If f is continuous on [a, b], then

${\int }_a^{b}f\left(x\right).dx={[\int f(x).dx]}_a^b$.