Definite and Indefinite Integrals

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

• antiderivative

• indefinite integral

• definite integral

• derivative

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  $\int_0^2\left(x^2-2x+1\right)\ dx$  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 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

Which of the following is incorrect?

•  $\int_b^af\left(x\right)\ dx=-\int_a^bf\left(x\right)\ dx$  

•  $\int_a^af\left(x\right)\ dx=0$  

•  $\int_a^bf\left(x\right)\ dx+\int_b^cf\left(x\right)\ dx$  
   $=\int_a^cf\left(x\right)\ dx$  
  

•  $\int_a^bf\left(x\right)\ dx-\int_b^cg\left(x\right)\ dx$  
   $=\int_a^c\left(f\left(x\right)-g\left(x\right)\right)\ dx$  
  

•  $\int_a^bf\left(x\right)\ dx-\int_a^bg\left(x\right)\ dx$  
   $=\int_a^b\left(f\left(x\right)-g\left(x\right)\right)\ dx$  
  

  ∫₁³ 5x² dx

• a. 43
• b. 130/3
• c. 40
• d. 19/3

∫₁² (3x² + 4x³)dx

• a. 24
• b. -8
• c. -10
• d. 22

• 12/3
• 6.5
• 5.5
• 11/3

Find the area under a curve defined by the equation 5x⁴+3x+7 between the x values 0 and 4.

• 1200
• 1/12
• 1134
• 1076