In how many ways can nine books be arranged on a shelf if the three math books must be placed next to each other?

Use the Fundamental Theorem of Calculus to find the area of the region between the graph of the function x5 + 8x4 + 2x2 + 5x + 1

belka [17]

Answer:

The area of the region between the graph of the given function and the x-axis = 25,351 units²

Step-by-step explanation:

Given x⁵ + 8 x⁴ + 2 x² + 5 x + 15

If 'f' is a continuous on [a ,b] then the function

$F(x) = \int\limits^a_b {f(x)} \, dx$

By using integration formula

$\int{x^n} \, dx = \frac{x^{n+1} }{n+1} +c$

Given x⁵ + 8 x⁴ + 2 x² + 5 x + 15 in the interval [-6,6]

$\int\limits^6_^-6} (x^{5} + 8 x^{4} + 2 x^{2} + 5 x + 15) )dx$

On integration , we get

= $(\frac{x^{6} }{6} + \frac{8 x^{5} }{5} + 2 \frac{x^{3} }{3} +\frac{5 x^{2} }{2} + 15 x)^{6} _{-6}$

$F(x) = \int\limits^a_b {f(x)} \, dx = F(b) -F(a)$

= $(\frac{6^{6} }{6} + \frac{8 6^{5} }{5} + 2 \frac{6^{3} }{3} +\frac{5 6^{2} }{2} + 15X 6) - ((\frac{(-6)^{6} }{6} + \frac{8 (-6)^{5} }{5} + 2 \frac{(-6)^{3} }{3} +\frac{5 (-6)^{2} }{2} + 15 (-6))$

After simplification and cancellation we get

= $\frac{2 X 8 X (6)^{5} }{5} + \frac{2 X 2 X (6)^3}{3} + 2 X 15 X 6$

on calculation , we get

= $\frac{124,416}{5} + \frac{864}{3} + 180$

On L.C.M 15

= $\frac{124,416 X 3 + 864 X 5 + 180 X 15}{15}$

= 25 351.2 units²

Conclusion:-

The area of the region between the graph of the given function and the x-axis = 25,351 units²

6 0

2 years ago

Solve for m and n using the special right triangles

Irina18 [472]

Since this is a 45-45-90 triangle, n = 6 and m = 6√3.

I hope this helps!

8 0

3 years ago

Given the isosceles trapezoid below, find AG. Show all the work

Dmitry [639]

We know the width of the rectangle in the middle of the trapezoid is 24 (from the top of the image), so we can subtract that from the bottom width of the trapezoid to get the combined length of the bottom of both triangles.

$40 - 24 = 16$

Since this is an isosceles trapezoid, both triangle bases are the same length, so we can cut this value in half to get the length of $GT$ and $EF$

$\frac{16}{2} = 8$

Finally, we can use the Pythagorean Theorem to find the length of $AG$ :

$AG^{2} + GT^{2} = AT^{2}$

$AG^{2} + 8^{2} = 10^{2}$

$AG^{2} + 64 = 100$

$AG^{2} = 36$

$AG = 6$

8 0

2 years ago

The perimeter of a rectangle is 62m. find the length of its sides if it is known that the area of the rectangle is 1980000 cm².

ss7ja [257]

The perimeter of a rectangle is 62m. find the length of its sides if it is known that the area of the rectangle is 1980000 cm².

6 0

2 years ago

Read 2 more answers

Two forest service watchtowers spot a fire in the distance. The fire is 27 miles from Tower 2. About how far apart are the two w

Reptile [31]

Given that the two watchtowers spotted fire in the distance and watch tower 2 was 27 miles from the fire. Assuming that the fire is in the middle of both watch towers, then the distance between the two watch towers will be given by:
distance=[distance of fire from watch tower 1]+[distance of fire from watch tower 2]
=27+27
=53 miles

8 0

3 years ago