Definite Integral Recap

Multiple Choice

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

• antiderivative

• indefinite integral

• definite integral

• derivative

Multiple Choice

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!

Multiple Choice

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

• positive

• negative

• zero

Multiple Choice

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

• positive

• negative

• zero

Multiple Choice

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

• positive

• negative

• zero

Multiple Choice

Evaluate the definite integral.

• 3
• 9/2
• 9
• Not possible

Multiple Choice

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

Multiple Choice

$$\int_0^{\frac{\pi}{2}}\cos\left(x\right)\ dx$$

• -1

• 1

• $$\pi$$  

• $$\frac{\pi}{2}$$  

Multiple Choice

Using the areas of each region given
$$\int_a^cf\left(x\right)=$$  

• 20

• -7

• -10

• -4

Notes

Multiple Choice

Which of the following is incorrect?

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

• $$\int_a^af\left(x\right)\ dx=0$$  

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

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

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

Multiple Choice

$$\int_1^4\ \left|x\ -\ 3\right|\ dx$$  =

• $$\frac{-5}{2}$$  

• 6

• $$\frac{-3}{2}$$  

• $$\frac{5}{2}$$  

Multiple Choice

$$\int_{-3}^5\ \left|x^2+7x\ -\ 18\right|\ dx$$  =

• 154.33

• -37.33

• 161

• 52.67

Multiple Choice

If $$\int_2^{10}f\left(x\right)dx=-6$$  
Find the value of  $$\int_2^{10}\ \frac{1}{3}f\left(x\right)dx$$  

• 2

• 18

• - 2

• - 18

Multiple Choice

If $$\int_2^{10}f\left(x\right)dx=-6$$  
Find the value of  $$\int_{10}^2f\left(x\right)dx$$  

• - 6

• 6

• 2

• 10

Multiple Choice

$$\int_{\frac{\pi}{6}}^{\frac{\pi}{3}}\left(1+\sin3t\right)\left(\cos3t\right)dt=$$  

• 0

• 1/2

• -1/2

• 1

Multiple Choice

$$\int_{-4}^4\left|x+2\right|dx=$$  

• 40

• 20

• 10

• Not possible

Multiple Choice

$$\int_0^{\pi}24\sin\left(6x\right)\cos\left(6x\right)dx=$$  

• -8/3

• -4/3

• 0

• 4/3

Multiple Choice

$$\int_{-1}^1\left(2x-1\right)\left(2x+1\right)dx=$$  

• 3

• $$-\frac{2}{3}$$  

• $$\frac{2}{3}$$  

• $$-1.5$$  

Multiple Choice

$$\int_1^2\left(2x+k\right)dx=8.$$  

$$\ k=?$$  

• $$4$$  

• $$5$$  

• $$-3$$  

• $$-2$$  

Multiple Choice

$$\int_{-1}^a\left(-3x+4\right)dx=\frac{49}{8}.$$  

$$\ a=?$$  

• $$1$$  

• $$2.5$$  

• $$-3$$  

• $$4$$  

Multiple Choice

If $$\int_{-4}^3f\left(x\right)dx=9,\ \ \int_3^5f\left(x\right)dx=-11,\ \ \&\ \ \int_{-4}^3h\left(x\right)dx=14$$  , then evaluate $$\int_5^3f\left(x\right)dx$$   if possible.

• -11

• 11

• 9

• not enough information to determine

Multiple Choice

If $$\int_{-4}^3f\left(x\right)dx=9,\ \ \int_3^5f\left(x\right)dx=-11,\ \ \&\ \ \int_{-4}^3h\left(x\right)dx=14$$  , then evaluate $$\int_{-4}^5f\left(x\right)dx$$   if possible.

• -11

• -2

• 20

• not enough information to determine

Multiple Choice

If $$\int_{-4}^3f\left(x\right)dx=9,\ \ \int_3^5f\left(x\right)dx=-11,\ \ \&\ \ \int_{-4}^3h\left(x\right)dx=14$$  , then evaluate $$\int_{-4}^3\left[f\left(x\right)-h\left(x\right)\right]dx$$   if possible.

• -5

• 5

• 23

• not enough information to determine

Multiple Choice

If $$\int_{-4}^3f\left(x\right)dx=9,\ \ \int_3^5f\left(x\right)dx=-11,\ \ \&\ \ \int_{-4}^3h\left(x\right)dx=14$$  , then evaluate $$\int_5^3-f\left(x\right)dx$$   if possible.

• -11

• 11

• 9

• not enough information to determine

Multiple Choice

If $$\int_{-4}^3f\left(x\right)dx=9,\ \ \int_3^5f\left(x\right)dx=-11,\ \ \&\ \ \int_{-4}^3h\left(x\right)dx=14$$  , then evaluate $$\int_3^3f\left(x\right)dx$$   if possible.

• -2

• 0

• 20

• not enough information to determine

Multiple Choice

Which of the following is equivalent to:

$$\int_0^3f\left(x\right)dx+\int_3^7f\left(x\right)dx$$  

• 7

• $$\int_0^7f\left(x\right)dx$$  

• $$\int_7^0f\left(x\right)dx$$  

• None of the following are equivalent to the given

Multiple Choice

The demand function, $$D(x)=10-x^2$$   and the supply function. $$S\left(x\right)=x^2+x$$   ,  where y represent the price in RM. Find the market equilibrium point

• $$x=2,y=6$$  

• RM5.33

• $$\left(-\frac{5}{2},6\right)$$  

• $$\left(2,6\right)$$  

Multiple Choice

Given the marginal cost function,   $$C'\left(x\right)=6x^2-3x+5$$  and the fixed cost is 8. Find the cost function.

• $$C\left(x\right)=2x^3-\frac{3}{2}x^2+$$  

  $$5x$$  

• $$C\left(x\right)=2x^3-\frac{3}{2}x^2+$$

  $$5x+8$$  

• $$C\left(x\right)=12x-3$$

• $$C\left(x\right)=2x^3-\frac{3}{2}x^2+$$

  $$5x+k$$  

Multiple Choice

Calculate the area for the shaded region on the function $$y=x^3-3x^2+2x$$  

• 1.4 units²

• 0.5 units²

• 0.56 units²

• 2.67 units²

• 0.33 units²

Multiple Choice

Find the area of the region bounded by the curves  $$y=x^2-5x$$    and  $$y=25-x^2$$  .

• $$140\ \frac{6}{8}$$  

• $$\frac{1125}{8}$$  

• $$140$$  

• $$\frac{1120}{8}$$