Definite Integrals

Definite integrals: Use the Fundamental Theorem of Calculus to find the exact values of each of the definite integrals that follow. Sketch the areas described by these definite integrals to determine whether your answers are reasonable. 

${\int }_{-1}^1\frac{x}{1+x^2}dx$

Definite integrals: Use the Fundamental Theorem of Calculus to find the exact values of each of the definite integrals that follow. Sketch the areas described by these definite integrals to determine whether your answers are reasonable.

${\int }_{-1}^1\frac{-1}{1+x^2}dx$

The signed area between the graph of  $f\left(x\right)=3x^2-7x+2$and the $x$-axis on $\left[0,4\right].$

The signed area between the graph of $f\left(x\right)=3x^2-7x+2$ and the x-axis on $[0,4].$

Calculating areas and average values: Express each of the following areas or average values with a definite integral, and then use definite integral formulas to compute the exact value of the area or average value. 

The absolute area between the graph of $f\left(x\right)=3x^2-7x+2$ and the $x$- axis is on $\left[0,4\right].$

The average value of the function $f\left(x\right)=\sin 2x,$on $\left[0,\mathrm{\pi }\right].$

The average value of the function $f\left(x\right)=\sin 2x on [0,\mathrm{\pi }].$

Calculate areas and average values : Express each of the following areas or average values with a definite integral, and then use definite integral formulas to compute the exact value of the area or average value.  

The area between the graphs of $f\left(x\right)=4-x^2.$ and $1-2x$ on $\left[-4,4\right].$ 

The area between the graphs of $f\left(x\right)=4-x^2 and 1-2x on [-4,4].$

Combining derivatives and integrals: Simplify each of the following as much as possible. 

$\frac{d}{dx}\int x^{3}dx$

Combining derivatives and integrals: Simplify each of the following as much as possible. 

$\frac{d}{dx}{\int }_0^x t^{3}dt$

Combining derivatives and integrals: Simplify each of the following as much as possible:

${\int }_{-2}^x\frac{d}{dx}\left(\ln \right(x^2+1\left)\right)dx$

Combining derivatives and integrals: Simplify each of the following as much as possible:

$\frac{d}{dx}{\int }_0^3{e}^{-t^2}dt$

Combining derivatives and integrals: Simplify each of the following as much as possible: 

$\frac{d}{dx}{\int }_0^x{e}^{-t^2}dt$

Combining derivatives and integrals: Simplify each of the following as much as possible: 

$\frac{d}{dx}{\int }_0^{\ln x}{\sin }^{3}tdt$

Combining derivatives and integrals: Simplify each of the following as much as possible: 

$\frac{d}{dx}{\int }_x^3{\sin }^{3}t dt$

Is integration the opposite of differentiation? In what sense do derivatives “undo” integrals? In what sense do integrals “undo” derivatives? In what sense do they not? In your answer, be sure to consider indefinite integrals, definite integrals, and accumulation functions defined by integrals. 

The Fundamental Theorem of Calculus: Why is the Fundamental Theorem of Calculus so fundamental? What does it allow us to calculate, and what concepts does it relate? Give an overview outline of the proof of this important theorem 

The area under a velocity curve: Return to the very start of this chapter, and review the discussion of driving down a straight road with stoplights. Describe in your own words the relationship velocity, distance, and accumulation functions illustrated in that discussion. Then use what you know from the material you learned in this chapter to calculate the exact distance travelled in that situation. 

Is integration the opposite of differentiation? In what sense do derivatives “undo” integrals? In what sense do integrals “undo” derivatives? In what sense do they not? In your answer, be sure to consider indefinite integrals, definite integrals, and accumulation functions defined by integrals 

The Fundamental Theorem of Calculus: Why is the Fundamental Theorem of Calculus so fundamental? What does it allow us to calculate, and what concepts does it relate? Give an overview outline of the proof of this important theorem. 

Defining logarithms with integrals: In this chapter we defined the natural logarithm function as the accumulation integral 

$\ln x={\int }_0^x \frac{1}{t}dt$

(a) Use the graph of $y=\frac{1}{x}$ and this definition to describe the graphical features of $y=\ln x.$

(b) Given this definition of $\ln x,$ how would we define the natural exponential function $e^x$? Why is this a better definition for $e^x$ than the one we introduced in Definition $1.25$?