Definite integrals and its properties

The area under a curve is calculated using which mathematical concept?

• antiderivative

• indefinite integral

• definite integral

• derivative

The area under a curve is also known as the ... ?

• physical area

• algebraic area

• integral

• antiderivative

The fundamental theorem of calculus is basically:

• calculating the area under a curve

•  $\int_a^b\ f\left(x\right)\ dx=F\left(b\right)-F\left(a\right)$ 

• integrating a function, with no +C

• using rectangles to approximate the area under a curve

• we didn't learn this!

The result of  $\int_1^3\left(x-2\right)\ dx$  is:

• positive

• negative

• zero

Tags

CCSS.HSA.SSE.A.1

The result of  $\int_0^2\left(x^2-2x+1\right)\ dx$  is:

• positive

• negative

• zero

Tags

CCSS.HSA.APR.A.1

CCSS.HSA.REI.B.4

CCSS.HSA.SSE.A.1

The result of  $\int_0^2\left(x^3-2x^2-x+1\right)\ dx$  is:

• positive

• negative

• zero

The shaded area in the figure is represented by which of the following integral expressions?

•  $\int_{-1}^2f\left(x\right)\ dx$  

•  $-\int_2^{-1}f\left(x\right)\ dx$  

•  $-\int_{-1}^2f\left(x\right)\ dx$  

•  $\int_{-2}^1f\left(x\right)\ dx$  

The shaded area shown in the figure can be represented by which of the following integral expressions?

•  $\int_0^3f\left(x\right)\ dx$  

•  $-\int_0^1f\left(x\right)\ dx+\int_1^3f\left(x\right)\ dx$  

•  $\int_0^1f\left(x\right)\ dx+\int_1^3f\left(x\right)\ dx$  

•  $\int_1^0f\left(x\right)\ dx+\int_1^3f\left(x\right)\ dx$  

Considering the area shown in the diagram, which of the following statements is not true?

• The algebraic area could be negative.

• The physical area is positive.

• Both the physical area and algebraic area could be negative.

• Both the physical area and algebraic area could be positive.

The area under a curve can be represented by  $\int_a^bf\left(x\right)\ dx$  only if:

•  $f\left(x\right)$  is positive

•  $f\left(x\right)$  is continuous

•  $f\left(a\right)$  exists

•  $f\left(b\right)$  exists