{"id":1368,"date":"2024-09-23T21:05:07","date_gmt":"2024-09-23T21:05:07","guid":{"rendered":"https:\/\/loudounmathtutoring.org\/?page_id=1368"},"modified":"2026-02-12T00:31:25","modified_gmt":"2026-02-12T00:31:25","slug":"teaching-aids-calculus-integration","status":"publish","type":"page","link":"https:\/\/loudounmathtutoring.org\/?page_id=1368","title":{"rendered":"Teaching Aids &#8211; Calculus (Integration)"},"content":{"rendered":"\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><a href=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/1stFundTheoremCalc.png\"><img loading=\"lazy\" decoding=\"async\" width=\"1024\" height=\"511\" src=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/1stFundTheoremCalc.png\" alt=\"\" class=\"wp-image-1369\" style=\"width:543px;height:auto\" srcset=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/1stFundTheoremCalc.png 1024w, https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/1stFundTheoremCalc-300x150.png 300w, https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/1stFundTheoremCalc-768x383.png 768w, https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/1stFundTheoremCalc-960x480.png 960w\" sizes=\"auto, (max-width: 1024px) 100vw, 1024px\" \/><\/a><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><a href=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/2ndFundTheorem.png\"><img loading=\"lazy\" decoding=\"async\" width=\"308\" height=\"163\" src=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/2ndFundTheorem.png\" alt=\"\" class=\"wp-image-1370\" style=\"width:406px;height:auto\" srcset=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/2ndFundTheorem.png 308w, https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/2ndFundTheorem-300x159.png 300w\" sizes=\"auto, (max-width: 308px) 100vw, 308px\" \/><\/a><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/DerivativeVsIntegration.gif\"><img loading=\"lazy\" decoding=\"async\" width=\"537\" height=\"727\" src=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/DerivativeVsIntegration.gif\" alt=\"\" class=\"wp-image-1371\"\/><\/a><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><a href=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/some-properties-of-definite-integral-formula-1619752195.png\"><img loading=\"lazy\" decoding=\"async\" width=\"885\" height=\"711\" src=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/some-properties-of-definite-integral-formula-1619752195.png\" alt=\"\" class=\"wp-image-1372\" style=\"width:662px;height:auto\" srcset=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/some-properties-of-definite-integral-formula-1619752195.png 885w, https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/some-properties-of-definite-integral-formula-1619752195-300x241.png 300w, https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/some-properties-of-definite-integral-formula-1619752195-768x617.png 768w\" sizes=\"auto, (max-width: 885px) 100vw, 885px\" \/><\/a><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><a href=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/integration-by-parts.png\"><img loading=\"lazy\" decoding=\"async\" width=\"398\" height=\"350\" src=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/integration-by-parts.png\" alt=\"\" class=\"wp-image-1373\" style=\"width:469px;height:auto\" srcset=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/integration-by-parts.png 398w, https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/integration-by-parts-300x264.png 300w\" sizes=\"auto, (max-width: 398px) 100vw, 398px\" \/><\/a><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/CommonIntegrals.png\"><img loading=\"lazy\" decoding=\"async\" width=\"512\" height=\"288\" src=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/CommonIntegrals.png\" alt=\"\" class=\"wp-image-1374\" srcset=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/CommonIntegrals.png 512w, https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/CommonIntegrals-300x169.png 300w\" sizes=\"auto, (max-width: 512px) 100vw, 512px\" \/><\/a><\/figure>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<figure class=\"wp-block-image size-full\"><a href=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/calculus-ab-bc-first-pic.png\"><img loading=\"lazy\" decoding=\"async\" width=\"869\" height=\"533\" src=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/calculus-ab-bc-first-pic.png\" alt=\"\" class=\"wp-image-1375\" srcset=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/calculus-ab-bc-first-pic.png 869w, https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/calculus-ab-bc-first-pic-300x184.png 300w, https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2024\/09\/calculus-ab-bc-first-pic-768x471.png 768w\" sizes=\"auto, (max-width: 869px) 100vw, 869px\" \/><\/a><\/figure>\n\n\n\n<p><strong>Riemann Sums:<\/strong><\/p>\n\n\n\n<hr class=\"wp-block-separator has-alpha-channel-opacity\"\/>\n\n\n\n<figure class=\"wp-block-gallery has-nested-images columns-default is-cropped wp-block-gallery-1 is-layout-flex wp-block-gallery-is-layout-flex\">\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/Left_Riemann_Sum.png\"><img loading=\"lazy\" decoding=\"async\" width=\"791\" height=\"721\" data-id=\"1711\" src=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/Left_Riemann_Sum.png\" alt=\"\" class=\"wp-image-1711\" srcset=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/Left_Riemann_Sum.png 791w, https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/Left_Riemann_Sum-300x273.png 300w, https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/Left_Riemann_Sum-768x700.png 768w\" sizes=\"auto, (max-width: 791px) 100vw, 791px\" \/><\/a><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/Midpoint_Riemann_Sum.png\"><img loading=\"lazy\" decoding=\"async\" width=\"654\" height=\"580\" data-id=\"1709\" src=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/Midpoint_Riemann_Sum.png\" alt=\"\" class=\"wp-image-1709\" srcset=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/Midpoint_Riemann_Sum.png 654w, https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/Midpoint_Riemann_Sum-300x266.png 300w\" sizes=\"auto, (max-width: 654px) 100vw, 654px\" \/><\/a><\/figure>\n\n\n\n<figure class=\"wp-block-image size-large\"><a href=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/Right_Riemann_Sum.png\"><img loading=\"lazy\" decoding=\"async\" width=\"670\" height=\"460\" data-id=\"1710\" src=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/Right_Riemann_Sum.png\" alt=\"\" class=\"wp-image-1710\" srcset=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/Right_Riemann_Sum.png 670w, https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/Right_Riemann_Sum-300x206.png 300w\" sizes=\"auto, (max-width: 670px) 100vw, 670px\" \/><\/a><\/figure>\n<\/figure>\n\n\n\n<p><strong>Trapezoidal Riemann Sum:<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><a href=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/Trapezoidal_Riemann_Sum-1.png\"><img loading=\"lazy\" decoding=\"async\" width=\"620\" height=\"435\" src=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/Trapezoidal_Riemann_Sum-1.png\" alt=\"\" class=\"wp-image-1720\" style=\"width:320px;height:auto\" srcset=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/Trapezoidal_Riemann_Sum-1.png 620w, https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/Trapezoidal_Riemann_Sum-1-300x210.png 300w\" sizes=\"auto, (max-width: 620px) 100vw, 620px\" \/><\/a><\/figure>\n\n\n\n<p><strong>Area Under Between Two Curves:<\/strong><\/p>\n\n\n\n<figure class=\"wp-block-image size-full is-resized\"><a href=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/AreaUnderCurve.png\"><img loading=\"lazy\" decoding=\"async\" width=\"867\" height=\"941\" src=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/AreaUnderCurve.png\" alt=\"\" class=\"wp-image-1715\" style=\"width:615px;height:auto\" srcset=\"https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/AreaUnderCurve.png 867w, https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/AreaUnderCurve-276x300.png 276w, https:\/\/loudounmathtutoring.org\/wp-content\/uploads\/2026\/02\/AreaUnderCurve-768x834.png 768w\" sizes=\"auto, (max-width: 867px) 100vw, 867px\" \/><\/a><\/figure>\n\n\n\n<p><\/p>\n","protected":false},"excerpt":{"rendered":"<p>Riemann Sums: Trapezoidal Riemann Sum: Area Under Between Two Curves:<\/p>\n","protected":false},"author":1,"featured_media":0,"parent":0,"menu_order":0,"comment_status":"closed","ping_status":"closed","template":"","meta":{"footnotes":""},"class_list":["post-1368","page","type-page","status-publish","hentry","post"],"_links":{"self":[{"href":"https:\/\/loudounmathtutoring.org\/index.php?rest_route=\/wp\/v2\/pages\/1368","targetHints":{"allow":["GET"]}}],"collection":[{"href":"https:\/\/loudounmathtutoring.org\/index.php?rest_route=\/wp\/v2\/pages"}],"about":[{"href":"https:\/\/loudounmathtutoring.org\/index.php?rest_route=\/wp\/v2\/types\/page"}],"author":[{"embeddable":true,"href":"https:\/\/loudounmathtutoring.org\/index.php?rest_route=\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/loudounmathtutoring.org\/index.php?rest_route=%2Fwp%2Fv2%2Fcomments&post=1368"}],"version-history":[{"count":15,"href":"https:\/\/loudounmathtutoring.org\/index.php?rest_route=\/wp\/v2\/pages\/1368\/revisions"}],"predecessor-version":[{"id":1724,"href":"https:\/\/loudounmathtutoring.org\/index.php?rest_route=\/wp\/v2\/pages\/1368\/revisions\/1724"}],"wp:attachment":[{"href":"https:\/\/loudounmathtutoring.org\/index.php?rest_route=%2Fwp%2Fv2%2Fmedia&parent=1368"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}