Question

Area functions for the same linear function Let ƒ(t) = t and consider the two area functions A(𝓍) = ∫₀ˣ ƒ(t) dt and F(𝓍) = ∫₂ˣ ƒ(t) dt .
(b) Evaluate F(4) and F(6). Then use geometry to find an expression for F (𝓍) , for 𝓍 ≥ 2.

Accepted Answer

Step 1: Understand the problem. We are given a linear function ƒ(t) = t and two area functions: A(𝓍) = ∫₀ˣ ƒ(t) dt and F(𝓍) = ∫₂ˣ ƒ(t) dt. The goal is to evaluate F(4) and F(6), and then derive a general expression for F(𝓍) using geometry for 𝓍 ≥ 2. Step 2: Recall the geometric interpretation of definite integrals. The integral ∫₂ˣ ƒ(t) dt represents the area under the curve ƒ(t) = t from t = 2 to t = 𝓍. Since ƒ(t) = t is a linear function, the graph is a straight line, and the area can be calculated as the area of a trapezoid or triangle. Step 3: Evaluate F(4). To find F(4), compute the definite integral ∫₂⁴ t dt. This represents the area under the line ƒ(t) = t from t = 2 to t = 4. Use the formula for the definite integral of t, ∫ t dt = (1/2)t², and evaluate it at the bounds 2 and 4. Step 4: Evaluate F(6). Similarly, compute the definite integral ∫₂⁶ t dt. This represents the area under the line ƒ(t) = t from t = 2 to t = 6. Again, use the formula for the definite integral of t, ∫ t dt = (1/2)t², and evaluate it at the bounds 2 and 6. Step 5: Derive a general expression for F(𝓍) for 𝓍 ≥ 2. Using geometry, note that the area under the line ƒ(t) = t from t = 2 to t = 𝓍 forms a trapezoid or triangle. The formula for the area of a trapezoid is (1/2)(base1 + base2) × height. Alternatively, use the definite integral formula ∫₂ˣ t dt = (1/2)𝓍² - (1/2)(2²) to express F(𝓍).

Area functions for the same linear function Let ƒ(t) = t and consider the two area functions A(𝓍) = ∫₀ˣ ƒ(t) dt and F(𝓍) = ∫₂ˣ ƒ(t) dt .
(b) Evaluate F(4) and F(6). Then use geometry to find an expression for F (𝓍) , for 𝓍 ≥ 2.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Definite Integral

Fundamental Theorem of Calculus

Geometric Interpretation of Integrals

Area functions for the same linear function Let ƒ(t) = t and consider the two area functions A(𝓍) = ∫₀ˣ ƒ(t) dt and F(𝓍) = ∫₂ˣ ƒ(t) dt .(b) Evaluate F(4) and F(6). Then use geometry to find an expression for F (𝓍) , for 𝓍 ≥ 2.

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

Definite Integral

Fundamental Theorem of Calculus

Geometric Interpretation of Integrals

Area functions for the same linear function Let ƒ(t) = t and consider the two area functions A(𝓍) = ∫₀ˣ ƒ(t) dt and F(𝓍) = ∫₂ˣ ƒ(t) dt .
(b) Evaluate F(4) and F(6). Then use geometry to find an expression for F (𝓍) , for 𝓍 ≥ 2.