Hi there!
Our interval is from 0 to 3, with 6 intervals. Thus:
3 ÷ 6 = 0.5, which is our width for each rectangle.
Since n = 6 and we are doing a right-riemann sum, the points we will be plugging in are:
0.5, 1, 1.5, 2, 2.5, 3
Evaluate:
(0.5 · f(0.5)) + (0.5 · f(1)) + (0.5 · f(1.5)) + (0.5 · f(2)) + (0.5 · f(2.5)) + (0.5 · f(3)) =
Simplify:
0.5( -2.75 + (-3) + (-.75) + 4 + 11.25 + 21) = 14.875
Answer:
General Formulas and Concepts:
<u>Pre-Algebra</u>
Order of Operations: BPEMDAS
Equality Properties
<u>Algebra I</u>
<u>Calculus</u>
Area - Integrals
Integration Rule [Reverse Power Rule]:
Integration Rule [Fundamental Theorem of Calculus 1]:
Integration Property [Addition/Subtraction]:
Area of a Region Formula:
Step-by-step explanation:
*Note:
<em>Remember that for the Area of a Region, it is top function minus bottom function.</em>
<u />
<u>Step 1: Define</u>
f(x) = x²
g(x) = x⁶
Bounded (Partitioned) by x-axis
<u>Step 2: Identify Bounds of Integration</u>
<em>Find where the functions intersect (x-values) to determine the bounds of integration.</em>
Simply graph the functions to see where the functions intersect (See Graph Attachment).
Interval: [-1, 1]
Lower bound: -1
Upper Bound: 1
<u>Step 3: Find Area of Region</u>
<em>Integration</em>
Topic: AP Calculus AB/BC (Calculus I/II)
Unit: Area Under the Curve - Area of a Region (Integration)
Book: College Calculus 10e
