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Aloiza [94]
3 years ago
8

Evaluate the integral ∫2−1|x−1|dx

Mathematics
1 answer:
defon3 years ago
5 0

I think you might be referring to the definite integral,

\displaystyle \int_{-1}^2|x-1|\,\mathrm dx

Recall the definition of absolute value:

|x| = \begin{cases}x&\text{if }x\ge0\\-x&\text{if }x

Then |x-1|=x-1 if x\ge1, and |x-1|=1-x is x. So spliting up the integral at <em>x</em> = 1, we have

\displaystyle \int_{-1}^2|x-1|\,\mathrm dx = \int_{-1}^1(1-x)\,\mathrm dx + \int_1^2(x-1)\,\mathrm dx

The rest is simple:

\displaystyle \int_{-1}^2|x-1|\,\mathrm dx = \left(x-\dfrac{x^2}2\right)\bigg|_{-1}^1 + \left(\dfrac{x^2}2-x\right)\bigg|_1^2 \\\\ = \left(\left(1-\frac12\right)-\left(-1-\frac12\right)\right) + \left(\left(2-2\right)-\left(\frac12-1\right)\right) \\\\ = \boxed{\frac52}

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The lengths of all sides of a triangle are integers when measured in decimeters. One of the sides has length of 1 dm and another
hram777 [196]

Answer:

The perimeter is 7 dm

Step-by-step explanation:

we know that

The Triangle Inequality Theorem states that the sum of the lengths of any two sides of a triangle is greater than the length of the third side.

Let

c------> the length of the third side

case a)

1+c>3

c>2 dm

case b)

1+3> c

4 > c

rewrite

c<4 dm

The solution for c is the interval------> (2,4)

therefore

the value of c must be equal to

c=3 dm (because is an integer)

The perimeter is equal to

P=1+3+3=7 dm

7 0
3 years ago
What the answer asap
Elan Coil [88]
20 qt = 80 c is ur answer. Hope this helps
3 0
3 years ago
Read 2 more answers
chen is buying a ham. He wants to divide it into 6.5-ounce servings for 12 people. write and solve a division equation to find w
OLga [1]

Answer:

Amount of people to serve x Serving Size = Size of ham

12 x 6.5 = ?

=78

Step-by-step explanation:

Looking for size ham chen should buy = ?

Amount of people to serve = 12

Serving Size = 6.5-ounce

Equation=

Amount of people to serve x Serving Size = Size of ham

12 x 6.5 = ?

=78

8 0
3 years ago
8. Stan made 4 goals out of 6 attempts in soccer today. What fraction
lisabon 2012 [21]

Answer:

\frac{4}{6}

Step-by-step explanation:

8 0
3 years ago
The following prism is made up of 27 identical
lutik1710 [3]

The greatest possible surface area of the prism is 78 square units, it is obtained by removing center cube on each side of the prism.

Step-by-step explanation:

The given is,

                Prism is made up of 27 identical  cubes

Step:1

               Ref attachment,

               Let, surface area of one of cube = 1 square units

               Surface area of given prism,

               In the given diagram it have 9 cube sides in each side of prism.

                                  1 surface prism = 9 surface of cube

               Surface area of given prism = 6 × Surface of prism

                                                              = 6 × 9

                                                              = 54 square units          

Step:2 Check for alternative's

          For removing one cube on the edge of prism,

                      1 surface of prim = 9 surfaces of cube

          Surface area after removing cube on each side,

                                             = 6 × 11 = 66 square units

           For removing  cube on corner of prism,

                      1 surface of prim = 9 surfaces of cube

           Surface area after removing cube on corner,

                                             = 6 × 9 = 54 square units

          For removing center cube on each side,

                     1 surface of prim = 13 surfaces of cube

           Surface area after removing cube on corer on the prism,

                                                  = 6 × 13 = 78 square units

   Surface area after removing corner cube on prism = 78 square units

Result:

       The greatest possible surface area of the prism is 78 square units, it is obtained by removing center cube on each side of the prism.

4 0
3 years ago
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