Reeves John
10/14/2023 · Middle School
Approximate the area under the graph of \( f(x)=0.03 x^{4}-1.21 x^{2}+64 \) over the interval \( [2,10] \) by dividing the interval into 4 subintervals, Use the left endpoint of each subinterval. The area under the graph of \( f(x)=0.03 x^{4}-1.21 x^{2}+64 \) over the interval \( \{2,10] \) is approximately (simplify your answer. Type an integer or a decimal.)
UpStudy ThothAI Solution
Tutor-Verified Answer
Step-by-step Solution
To approximate the area under the graph of the function \( f(x) = 0.03x^4 - 1.21x^2 + 64 \) over the interval \([2, 10]\) by dividing the interval into 4 subintervals using the left endpoint of each subinterval, we can use the left Riemann sum formula.
The left Riemann sum formula is given by:
\[ \text{Area} \approx \Delta x \times \left( f(x_0) + f(x_1) + f(x_2) + f(x_3) \right) \]
where:
- \( \Delta x = \frac{b - a}{n} \) is the width of each subinterval,
- \( n \) is the number of subintervals,
- \( x_0, x_1, x_2, x_3 \) are the left endpoints of the subintervals.
Given:
- \( f(x) = 0.03x^4 - 1.21x^2 + 64 \)
- Interval \([2, 10]\)
- Number of subintervals \( n = 4 \)
First, let's calculate the width of each subinterval:
\[ \Delta x = \frac{10 - 2}{4} = \frac{8}{4} = 2 \]
Next, we need to find the left endpoints of the subintervals:
- \( x_0 = 2 \)
- \( x_1 = 2 + 2 = 4 \)
- \( x_2 = 4 + 2 = 6 \)
- \( x_3 = 6 + 2 = 8 \)
Now, we can calculate the area using the left Riemann sum formula:
\[ \text{Area} \approx 2 \times \left( f(2) + f(4) + f(6) + f(8) \right) \]
Let's calculate the values of \( f(2) \), \( f(4) \), \( f(6) \), and \( f(8) \) and then find the approximate area.
Calculate the value by following steps:
- step0: Calculate:
\(0.03\times 2^{4}-1.21\times 2^{2}+64\)
- step1: Multiply the numbers:
\(0.48-1.21\times 2^{2}+64\)
- step2: Multiply the numbers:
\(0.48-4.84+64\)
- step3: Calculate:
\(59.64\)
Calculate or simplify the expression \( 0.03*4^4 - 1.21*4^2 + 64 \).
Calculate the value by following steps:
- step0: Calculate:
\(0.03\times 4^{4}-1.21\times 4^{2}+64\)
- step1: Multiply the numbers:
\(7.68-1.21\times 4^{2}+64\)
- step2: Multiply the numbers:
\(7.68-19.36+64\)
- step3: Calculate:
\(52.32\)
Calculate or simplify the expression \( 0.03*6^4 - 1.21*6^2 + 64 \).
Calculate the value by following steps:
- step0: Calculate:
\(0.03\times 6^{4}-1.21\times 6^{2}+64\)
- step1: Multiply the numbers:
\(38.88-1.21\times 6^{2}+64\)
- step2: Multiply the numbers:
\(38.88-43.56+64\)
- step3: Calculate:
\(59.32\)
Calculate or simplify the expression \( 0.03*8^4 - 1.21*8^2 + 64 \).
Calculate the value by following steps:
- step0: Calculate:
\(0.03\times 8^{4}-1.21\times 8^{2}+64\)
- step1: Multiply the numbers:
\(122.88-1.21\times 8^{2}+64\)
- step2: Multiply the numbers:
\(122.88-77.44+64\)
- step3: Calculate:
\(109.44\)
Calculate or simplify the expression \( 2*(59.64 + 52.32 + 59.32 + 109.44) \).
Calculate the value by following steps:
- step0: Calculate:
\(2\left(59.64+52.32+59.32+109.44\right)\)
- step1: Add the numbers:
\(2\times 280.72\)
- step2: Multiply the numbers:
\(561.44\)
The approximate area under the graph of \( f(x) = 0.03x^4 - 1.21x^2 + 64 \) over the interval \([2, 10]\) by dividing the interval into 4 subintervals using the left endpoint of each subinterval is approximately 561.44 square units.
Quick Answer
The approximate area is 561.44 square units.
Answered by UpStudy AI and reviewed by a Professional Tutor
UpStudy ThothAI
Self-Developed and Ever-Improving
Thoth AI product is constantly being upgraded and optimized.
Covers All Major Subjects
Capable of handling homework in math, chemistry, biology, physics, and more.
Instant and Accurate
Provides immediate and precise solutions and guidance.
Try Now
Ask Tutors
Ask AI
10x
Fastest way to Get Answers & Solutions
By text
Enter your question here…
By image
Re-Upload
Submit