Answer:
1/5
Step-by-step explanation:
1 + 4 = 5
1/5
Answer:
8
Step-by-step explanation:
Two different approaches:
<u>Method 1</u>
Apply radical rule √(ab) = √a√b to simplify the radicals:
√98 = √(49 x 2) = √49√2 = 7√2
√50 = √(25 x 2) = √25√2 = 5√2
Therefore, (√98 - √50)² = (7√2 - 5√2)²
= (2√2)²
= 4 x 2
= 8
<u>Method 2</u>
Use the perfect square formula: (a - b)² = a² - 2ab + b²
where a = √98 and b = √50
So (√98 - √50)² = (√98)² - 2√98√50 + (√50)²
= 98 - 2√98√50 + 50
= 148 - 2√98√50
Apply radical rule √(ab) = √a√b to simplify radicals:
√98 = √(49 x 2) = √49√2 = 7√2
√50 = √(25 x 2) = √25√2 = 5√2
Therefore, 148 - 2√98√50 = 148 - (2 × 7√2 × 5√2)
= 148 - 140
= 8
Split up the integration interval into 4 subintervals:
![\left[0,\dfrac\pi8\right],\left[\dfrac\pi8,\dfrac\pi4\right],\left[\dfrac\pi4,\dfrac{3\pi}8\right],\left[\dfrac{3\pi}8,\dfrac\pi2\right]](https://tex.z-dn.net/?f=%5Cleft%5B0%2C%5Cdfrac%5Cpi8%5Cright%5D%2C%5Cleft%5B%5Cdfrac%5Cpi8%2C%5Cdfrac%5Cpi4%5Cright%5D%2C%5Cleft%5B%5Cdfrac%5Cpi4%2C%5Cdfrac%7B3%5Cpi%7D8%5Cright%5D%2C%5Cleft%5B%5Cdfrac%7B3%5Cpi%7D8%2C%5Cdfrac%5Cpi2%5Cright%5D)
The left and right endpoints of the 
-th subinterval, respectively, are
for 
, and the respective midpoints are
We approximate the (signed) area under the curve over each subinterval by
so that
We approximate the area for each subinterval by
so that
We first interpolate the integrand over each subinterval by a quadratic polynomial 
, where
so that
It so happens that the integral of reduces nicely to the form you're probably more familiar with,
Then the integral is approximately
Compare these to the actual value of the integral, 3. I've included plots of the approximations below.
